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.

Wave Propagation in Two-Dimensional Nonlinear Periodic Lattices

2009 ◽  
Author(s):  
Raj K. Narisetti ◽  
Massimo Ruzzene ◽  
Michael J. Leamy
Keyword(s):  
Wave Propagation ◽  
Linear Models ◽  
Periodic Structures ◽  
Harmonic Wave ◽  
Excitation Amplitude ◽  
Pass Band ◽  
Perturbation Approach ◽  
Nonlinear Stiffness ◽  
Two Dimensional ◽  
Low Amplitude

This paper investigates wave propagation in two-dimensional nonlinear periodic structures subject to point harmonic forcing. The infinite lattice is modeled as a springmass system consisting of linear and cubic-nonlinear stiffness. The effects of nonlinearity on harmonic wave propagation are analytically predicted using a novel perturbation approach. Response is characterized by group velocity contours (derived from phase-constant contours) functionally dependent on excitation amplitude and the nonlinear stiffness coefficients. Within the pass band there is a frequency band termed the “caustic band” where the response is characterized by the appearance of low amplitude regions or “dead zones.” For a two-dimensional lattice having asymmetric nonlinearity, it is shown that these caustic bands are dependent on the excitation amplitude, unlike in corresponding linear models. The analytical predictions obtained are verified via comparisons to responses generated using a time-domain simulation of a finite two-dimensional nonlinear lattice. Lastly, the study demonstrates amplitude-dependent wave beaming in two-dimensional nonlinear periodic structures.

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.”

A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Raj K. Narisetti ◽  
Michael J. Leamy ◽  
Massimo Ruzzene
Keyword(s):  
Wave Propagation ◽  
Numerical Simulations ◽  
Perturbation Analysis ◽  
Periodic Structures ◽  
Harmonic Wave ◽  
Dispersion Relations ◽  
Perturbation Approach ◽  
One Dimensional ◽  
Chain Unit ◽  
Unit Cells

Wave propagation in one-dimensional nonlinear periodic structures is investigated through a novel perturbation analysis and accompanying numerical simulations. Several chain unit cells are considered featuring a sequence of masses connected by linear and cubic springs. Approximate closed-form, first-order dispersion relations capture the effect of nonlinearities on harmonic wave propagation. These relationships document amplitude-dependent behavior to include tunable dispersion curves and cutoff frequencies, which shift with wave amplitude. Numerical simulations verify the dispersion relations obtained from the perturbation analysis. The simulation of an infinite domain is accomplished by employing viscous-based perfectly matched layers appended to the chain ends. Numerically estimated wavenumbers show good agreement with the perturbation predictions. Several example chain unit cells demonstrate the manner in which nonlinearities in periodic systems may be exploited to achieve amplitude-dependent dispersion properties for the design of tunable acoustic devices.

Thermoelastic Small-Amplitude Wave Propagation in Nonlinear Elastic Multilayers

2004 ◽  
Vol 9 (5) ◽  
pp. 555-568 ◽  
Author(s):  
Massimiliano Gei ◽  
Davide Bigoni ◽  
Giulia Franceschni
Keyword(s):  
Wave Propagation ◽  
Small Amplitude ◽  
Amplitude Wave ◽  
Nonlinear Elastic

DQFEM Analyses of Static and Dynamic Nonlinear Elastic-Plastic Problems Using a GSR-Based Accelerated Constant Stiffness Equilibrium Iteration Technique

2001 ◽  
Vol 123 (3) ◽  
pp. 310-317 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Finite Element ◽  
Numerical Technique ◽  
Time Integration ◽  
Differential Quadrature ◽  
Element Discretization ◽  
Constant Stiffness ◽  
Structural Problems ◽  
Nonlinear Structural ◽  
Numerical Time Integration ◽  
Nonlinear Elastic

An integrated numerical technique for static and dynamic nonlinear structural problems adopting the equilibrium iteration is proposed. The differential quadrature finite element method (DQFEM), which uses the differential quadrature (DQ) techniques to the finite element discretization, is used to analyze the static and dynamic nonlinear structural mechanics problems. Numerical time integration in conjunction with the use of equilibrium iteration is used to update the response history. The equilibrium iteration can be carried out by the accelerated iteration schemes. The global secant relaxation-based accelerated constant stiffness and diagonal stiffness-based predictor-corrector equilibrium iterations which are efficient and reliable are used for the numerical computations. Sample problems are analyzed. Numerical results demonstrate the algorithm.

scholarly journals Micro-Vibrations and Wave Propagation in Biperiodic Cylindrical Shells

2020 ◽  
Vol 22 (3) ◽  
pp. 789-808
Author(s):  
Barbara Tomczyk ◽  
Anna Litawska
Keyword(s):  
Wave Propagation ◽  
Periodic Structures ◽  
Cylindrical Shells ◽  
Combined Model ◽  
Asymptotic Models ◽  
Wave Propagation Problem ◽  
Modelling Procedure ◽  
Constant Coefficients ◽  
A Cell ◽  
Averaged Model

AbstractThe objects of consideration are thin linearly elastic Kirchhoff-Love-type circular cylindrical shells having a periodically microheterogeneous structure in circumferential and axial directions (biperiodic shells). The aim of this contribution is to study a certain long wave propagation problem related to micro-fluctuations of displacement field caused by a periodic structure of the shells. This micro-dynamic problem will be analysed in the framework of a certain mathematical averaged model derived by means of the combined modelling procedure. The combined modelling applied here includes two techniques: the asymptotic modelling procedure and a certain extended version of the known tolerance non-asymptotic modelling technique based on a new notion of weakly slowly-varying function. Both these procedures are conjugated with themselves under special conditions. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, governing equations of the averaged combined model have constant coefficients depending also on a cell size. It will be shown that the micro-periodic heterogeneity of the shells leads to exponential micro-vibrations and to exponential waves as well as to dispersion effects, which cannot be analysed in the framework of the asymptotic models commonly used for investigations of vibrations and wave propagation in the periodic structures.

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