A Perturbation Approach for Analyzing Dispersion and Group Velocities in Two-Dimensional Nonlinear Periodic Lattices

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
R. K. Narisetti ◽  
M. Ruzzene ◽  
M. J. Leamy
Keyword(s):  
Perturbation Technique ◽  
Equations Of Motion ◽  
Wave Dispersion ◽  
Perturbation Approach ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Nonlinear Lattices ◽  
Directivity Patterns ◽  
Group Velocities

The paper investigates wave dispersion in two-dimensional, weakly nonlinear periodic lattices. A perturbation approach, originally developed for one-dimensional systems and extended herein, allows for closed-form determination of the effects nonlinearities have on dispersion and group velocity. These expressions are used to identify amplitude-dependent bandgaps, and wave directivity in the anisotropic setting. The predictions from the perturbation technique are verified by numerically integrating the lattice equations of motion. For small amplitude waves, excellent agreement is documented for dispersion relationships and directivity patterns. Further, numerical simulations demonstrate that the response in anisotropic nonlinear lattices is characterized by amplitude-dependent “dead zones.”

The Asymptotic Stability of Weakly Perturbed Two Dimensional Hamiltonian Systems

2001 ◽  
Author(s):  
Ludwig Arnold ◽  
Peter Imkeller ◽  
N. Sri Namachchivaya
Keyword(s):  
Lyapunov Exponent ◽  
Equations Of Motion ◽  
Perturbation Approach ◽  
Two Dimensional ◽  
White Noise Process ◽  
System A ◽  
Additive White Noise ◽  
Small Dissipation ◽  
Small Intensity ◽  
Single Well

Abstract The purpose of this work is to obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of a single-well Kramers Oscillator driven by either a multiplicative or an additive white noise process. To this end, we consider the equations of motion as dissipative and noisy perturbations of a two-dimensional Hamiltonian system. A perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ε and dissipation ε the exponent grows proportional to ε1/3.

Weakly nonlinear oscillatory convection in a rotating fluid

1992 ◽  
Vol 436 (1896) ◽  
pp. 33-54 ◽  
Keyword(s):  
Standing Wave ◽  
Perturbation Technique ◽  
Vertical Axis ◽  
Travelling Wave ◽  
Base Flow ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Weakly Nonlinear ◽  
Wave Disturbances ◽  
The Stability

The problem of weakly nonlinear two- and three-dimensional oscillatory convection in the form of standing waves is studied for a horizontal layer of fluid heated from below and rotating about a vertical axis. The solutions to the nonlinear problem are determined by a perturbation technique and the stability of all the base flow solutions is investigated with respect to both standing wave and travelling wave disturbances. The results of the stability and the nonlinear analyses for various values of the rotation parameter τ and the Prandtl number P (0 ≼ P < 0.677) indicate that there is no subcritical instability and that all the base flow solutions are unstable. Disturbances with highest growth rates are found to be some particular disturbances superimposed on two-dimensional base flow. Particular standing wave disturbances parallel to two-dimensional base flow are the most unstable ones either for sufficiently small P or for intermediate values of P with τ below some critical value τ *. Travelling wave disturbances inclined at an angle of about 45° to the wave vector of two-dimensional base flow are the most unstable disturbances either for P sufficiently close to its upper limit or for intermediate values of P with τ ≽ τ *. The dependence on P and τ of the nonlinear effect on the frequency and of the heat flux are also discussed.

Nonlinear Oscillation of a Cylinder Containing a Flowing Fluid

1969 ◽  
Vol 91 (4) ◽  
pp. 1147-1155 ◽  
Author(s):  
A. L. Thurman ◽  
C. D. Mote
Keyword(s):  
Approximate Solution ◽  
Fluid Transport ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Practical Interest ◽  
Longitudinal Motion ◽  
Flowing Fluid ◽  
Weakly Nonlinear ◽  
Efficient Calculation

The fundamental and second periods of transverse oscillation of a cylinder containing a flowing fluid are theoretically determined for the approximate solution of two, coupled, nonlinear partial differential equations describing the transverse and longitudinal motion. The calculations indicate that the existence of the fluid transport velocity reduces all cylinder natural periods of oscillation and increases the relative importance of non-linear terms in the equations of motion. Accordingly, in many cases of practical interest the linear analysis is shown to be severely limited in its applicability. Curves are presented that will assist one to estimate both the accuracy of the linear period and the approximate nonlinear period in selected examples. A new approximate solution method is utilized that permits accurate and efficient calculation of the nonlinear period. This method can be applied to the period determination of additional cylindrical models not examined herein; the method appears to be semi-generally applicable to the periodic solution of weakly nonlinear systems.

Wave Propagation in Membrane-Based Nonlinear Periodic Structures

2011 ◽  
Author(s):  
Raj K. Narisetti ◽  
Massimo Ruzzene ◽  
Michael J. Leamy
Keyword(s):  
Wave Propagation ◽  
Periodic Structures ◽  
Ultrasonic Transducers ◽  
Perturbation Approach ◽  
Element Discretization ◽  
Weakly Nonlinear ◽  
Optical Modes ◽  
Amplitude Dependency ◽  
Nonlinear Elastic ◽  
Group Velocities

Wave propagation in a periodic structure, formed by membrane elements on nonlinear elastic supports, is studied using a finite element discretization of a single unit cell followed by a perturbation analysis. The study is motivated in part by the need to study the dynamic behavior of micro-machined ultrasonic transducers (CMUTs). The requisite small parameter in the system arises from the ratio of the membrane to flexible support stiffness. The perturbation approach recovers linear Bloch formalism at first order, and amplitude-dependent dispersion corrections at higher orders. The procedure is used to generate weakly nonlinear band diagrams, which can in turn be used to identify amplitude-dependent bandgaps and group velocities. These diagrams also reveal that the strongest amplitude dependency occurs in high-frequency optical modes. Ultimately, the predicted dispersion behavior will be useful in assessing inter-element coupling and identifying effective excitation strategies for actuating CMUTs.

scholarly journals Complete Determination of Relaxation Parameters From Two-Dimensional Raman Spectroscopy

1999 ◽  
Vol 19 (1-4) ◽  
pp. 109-116 ◽  
Author(s):  
Vladimir Chernyak ◽  
Andrei Piryatinski ◽  
Shaul Mukamel
Keyword(s):  
Equations Of Motion ◽  
Raman Signal ◽  
Two Dimensional ◽  
Echo Signal ◽  
Time Resolved ◽  
Vibrational Coherence ◽  
Relaxation Parameters ◽  
Complete Determination ◽  
Population Relaxation

Using the model of a weakly-anharmonic, underdamped oscillator coupled to a bath, we demonstrate that the 2D time-resolved Raman signal carries information about the population decay T1, the homogeneous dephasing T2, and the inhomogeneous dephasing T1 relaxation timescales. We distinguish between two projections of the 2D signal: first, the echo signal, which is stretched in the diagonal direction decays with T2, and the second which is stretched along one axis is related to the population relaxation and decays with T1. The width of both signals reflects T1. Equations of motion for vibrational coherence and population variables are employed in these calculations.

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