WFQEM-based perturbation approach and its applications in analyzing nonlinear periodic structures

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.

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Impact of non-linear resonators in periodic structures using a perturbation approach

Mechanical Systems and Signal Processing ◽

10.1016/j.ymssp.2019.106408 ◽

2020 ◽

Vol 135 ◽

pp. 106408 ◽

Cited By ~ 3

Author(s):

M.A. Campana ◽

M. Ouisse ◽

E. Sadoulet-Reboul ◽

M. Ruzzene ◽

S. Neild ◽

...

Keyword(s):

Periodic Structures ◽

Perturbation Approach ◽

Non Linear

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Wave Propagation in Membrane-Based Nonlinear Periodic Structures

Volume 1: 23rd Biennial Conference on Mechanical Vibration and Noise, Parts A and B ◽

10.1115/detc2011-48700 ◽

2011 ◽

Cited By ~ 1

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.

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Modal Analysis and Forced Response of Coupled Mistuned Cyclic Systems: A Singular Perturbation Approach

Volume 3B: General ◽

10.1115/93-gt-266 ◽

1993 ◽

Cited By ~ 1

Author(s):

G. S. Happawan ◽

A. K. Bajaj ◽

O. D. I. Nwokah

Keyword(s):

Singular Perturbation ◽

Periodic Structures ◽

Forced Response ◽

Perturbation Approach ◽

Cyclic Symmetry ◽

Specific System ◽

Strongly Coupled ◽

Modal Expansion ◽

Parameter Perturbations ◽

Bladed Disk Assembly

The effect of small deterministic parameter perturbations on the forced response of nearly periodic structures with cyclic symmetry has been investigated. The general theory developed herein is applicable to any nth order strongly coupled cyclic system with two arbitrary and independent variations in system parameters that destroy the cyclic symmetry. The specific system studied may be regarded as a simplified model of a strongly coupled bladed-disk assembly. Singular perturbation methods along with modal expansion analysis are applied to gain a physical insight into the effects of perturbations on the eigenvalues, the eigenvectors as well as the forced response amplitudes. The study shows that, under appropriate conditions, the splitting and veering of eigenvalues due to mistunings or small parameter variations increases the amplitude of vibration of some blades quite significantly than would be predicted on the basis of an analysis of the perfectly tuned system. Further, the modal bifurcations lead to uneven vibration amplitudes irrespective of the stiffness of the coupling springs. The variation in blade amplitudes are also found to be strongly dependent on the type of engine order excitation for the same set of mistuning parameter.

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

Journal of Vibration and Acoustics ◽

10.1115/1.4000775 ◽

2010 ◽

Vol 132 (3) ◽

Cited By ~ 105

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.

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