Analysis of Damped Bloch Waves by the Rayleigh Perturbation Method

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
A. Srikantha Phani ◽  
Mahmoud I. Hussein
Keyword(s):  
Perturbation Method ◽  
State Space ◽  
Phononic Crystal ◽  
Quantitative Agreement ◽  
Unit Cell ◽  
Bloch Waves ◽  
Finite Structures ◽  
Complex Eigenvalues ◽  
Mass Chain ◽  
Computationally Intensive

Bloch waves in viscously damped periodic material and structural systems are analyzed using a perturbation method originally developed by Rayleigh for vibration analysis of finite structures. The extended method, called the Bloch–Rayleigh perturbation method here, utilizes the Bloch waves of an undamped unit cell as basis functions to provide approximate closed-form expressions for the complex eigenvalues and eigenvectors of the damped unit cell. In doing so, we circumvent the solution of a quadratic Bloch eigenvalue problem and subsequent computationally intensive transformation to first order/state-space form. Dispersion curves of a one-dimensional damped spring-mass chain and a two-dimensional phononic crystal with square inclusions are calculated using the state-space method and the proposed method. They are compared and found to be in excellent quantitative agreement for both proportional and nonproportional viscous damping models. The perturbation method is able to capture anomalous dispersion phenomena—branch overtaking, branch cut-on/cut-off, and frequency contour transformation—in parametric ranges where state-space formulations encounter numerical issues. Generalization to other linear nonviscous damping models is permissible.

Locally resonant effective phononic crystals for subwavelength vibration control of torsional cylindrical waves

2021 ◽  
pp. 1-30
Author(s):  
Ignacio Arretche ◽  
Kathryn Matlack
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Effective Properties ◽  
Equations Of Motion ◽  
Phononic Crystal ◽  
Phononic Crystals ◽  
Unit Cell ◽  
Periodic Coefficients ◽  
Bloch Theorem ◽  
Plane Wave Propagation

Abstract Locally resonant materials allow for wave propagation control in the sub-wavelength regime. Even though these materials do not need periodicity, they are usually designed as periodic systems since this allows for the application of the Bloch theorem and analysis of the entire system based on a single unit cell. However, geometries that are invariant to translation result in equations of motion with periodic coefficients only if we assume plane wave propagation. When wave fronts are cylindrical or spherical, a system realized through tessellation of a unit cell does not result in periodic coefficients and the Bloch theorem cannot be applied. Therefore, most studies of periodic locally resonant systems are limited to plane wave propagation. In this paper, we address this limitation by introducing a locally resonant effective phononic crystal composed of a radially-varying matrix with attached torsional resonators. This material is not geometrically periodic but exhibits effective periodicity, i.e. its equations of motion are invariant to radial translations, allowing the Bloch theorem to be applied to radially propagating torsional waves. We show that this material can be analyzed under the already developed framework for metamaterials. To show the importance of using an effectively periodic system, we compare its behavior to a system that is not effectively periodic but has geometric periodicity. We show considerable differences in transmission as well as in the negative effective properties of these two systems. Locally resonant effective phononic crystals open possibilities for subwavelength elastic wave control in the near field of sources.

scholarly journals Nonlinear Bloch waves and balance between hardening and softening dispersion

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180173 ◽  
Author(s):  
M. I. Hussein ◽  
R. Khajehtourian
Keyword(s):  
Dispersion Relation ◽  
Cell Size ◽  
Unit Cell ◽  
Homogeneous Layer ◽  
Nonlinear Dispersion ◽  
Elastic Band ◽  
Bloch Waves ◽  
Unit Cell Size ◽  
Nonlinear Dispersion Relation ◽  
Hardening And Softening

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

High Voltage Energy Harvesting From Embedded PVDF Harvester Inspired From Metamaterial Design

2019 ◽  
Author(s):  
Mohammadsadegh Saadatzi ◽  
Mohammad Nasser Saadatzi ◽  
Sourav Banerjee
Keyword(s):  
Energy Harvesting ◽  
Structural Design ◽  
Renewable Energy Sources ◽  
Phononic Crystal ◽  
Electrical Potential ◽  
Electrical Energy ◽  
Unit Cell ◽  
Energy Harvesters ◽  
Resonance Frequencies ◽  
Specific Material

Abstract In the current study, a novel multi-frequency, vibration-based Energy Harvester (EH) is proposed, numerically verified, and experimentally validated. The structural design of the proposed EH is inspired from an inner-ear, snail-shaped structure. In the past decade, scavenging power from environmental sources of vibration has attracted a lot of researchers to the field of energy harvesting. High demands for cleaner and renewable energy sources, limited sources of electrical energy, high depletion rates of nonrenewable sources of energy, and environmental concerns have urged researchers to investigate new structures called Metamaterial energy harvesters to harness electrical potential. The proposed EH is a metamaterial structure which has a Polyvinylidene Difluoride (PVDF) structure incapsulated in an aluminum frame and follows the physics of a mass-in-mass Phononic crystal structure. The PVDF snail-shaped structure is encapsulated inside a silicone matrix with a specific material property. This EH reacts to the environmental vibrations and the encapsulating silicone entraps the kinetic energy within its structure. The EH unit cell behaves as a negative mass in the vicinity of its resonance frequencies. In this paper, the dynamic behavior of the proposed EH is numerically modeled in COMSOL Multiphysics and, subsequently, validated experimentally using a unit cell fabricated in-house.

scholarly journals Highly Localized and Efficient Energy Harvesting in a Phononic Crystal Beam: Defect Placement and Experimental Validation

Crystals ◽  
2019 ◽  
Vol 9 (8) ◽  
pp. 391 ◽  
Author(s):  
Xu-Feng Lv ◽  
Xiang Fang ◽  
Zhi-Qiang Zhang ◽  
Zhi-Long Huang ◽  
Kuo-Chih Chuang
Keyword(s):  
Energy Harvesting ◽  
Point Defect ◽  
Polyvinylidene Fluoride ◽  
Experimental Validation ◽  
Flexural Rigidity ◽  
Phononic Crystal ◽  
Defect Mode ◽  
Unit Cell ◽  
Pvdf Film ◽  
Efficient Energy

We study energy harvesting in a binary phononic crystal (PC) beam at the defect mode. Specifically, we consider the placement of a mismatched unit cell related to the excitation point. The mismatched unit cell contains a perfect segment and a geometrically mismatched one with a lower flexural rigidity which serves as a point defect. We show that the strain in the defect PC beam is much larger than those in homogeneous beams with a defect segment. We suggest that the defect segment should be arranged in the first unit cell, but not directly connected to the excitation source, to achieve efficient less-attenuated localized energy harvesting. To harvest the energy, a polyvinylidene fluoride (PVDF) film is attached on top of the mismatched segment. Our numerical and experimental results indicate that the placement of the mismatched segment, which has not been addressed for PC beams under mechanical excitation, plays an important role in efficient energy harvesting based on the defect mode.

Exact Solution of Time History Response for Dynamic Systems With Arbitrary Viscous Damping Using Complex Modal Analysis

2007 ◽  
Author(s):  
Lonny L. Thompson ◽  
Manoj Kumar M. Chinnakonda
Keyword(s):  
Modal Analysis ◽  
State Space ◽  
Piecewise Linear ◽  
Time History ◽  
Quadrature Rules ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Complex Eigenvalues ◽  
Time History Response

A solution method for general, non-proportional damping time history response for piecewise linear loading is generalized to exact solutions which include piecewise quadratic loading. Comparisons are made to Trapezoidal and Simpson’s quadrature rules for approximating the time integral of the weighted generalized forcing function in the exact solution to the decoupled modal equations arising from state-space modal analysis of linear dynamic systems. Closed-form expressions for the weighting parameters in the quadrature formulas in terms of time-step size and complex eigenvalues are derived. The solution is obtained step-by-step from update formulas derived from the piecewise linear and quadratic interpolatory quadrature rules starting from the initial condition. An examination of error estimates for the different force interpolation methods shows convergence rates depend explicitly on the amount of damping in the system as measured by the real-part of the complex eigenvalues of the state-space modal equations and time-step size. Numerical results for a system with general, non-proportional damping, and driven by a continuous loading shows that for systems with light damping, update formulas for standard Trapezoidal and Simpson’s rule integration have comparable accuracy to the weighted piecewise linear and quadratic force interpolation update formulas, while for heavy damping, the update formulas from the weighted force interpolation quadrature rules are more accurate. Using a simple model representing a stiff system with general damping, we show that a two-step modal analysis using real-valued modal reduction followed by state-space modal analysis is shown to be an effective approach for rejecting spurious modes in the spatial discretization of a continuous system.

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