Analysis of Bloch’s Method and the Propagation Technique in Periodic Structures

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Farhad Farzbod ◽  
Michael J. Leamy
Keyword(s):  
Wave Propagation ◽  
Periodic Structures ◽  
Propagation Constant ◽  
Two Dimensional ◽  
Mathematical Basis ◽  
Translational Invariance ◽  
Nonlinear Media ◽  
Internal Forces ◽  
Propagation Technique ◽  
Rigorous Mathematical Analysis

Bloch analysis was originally developed by Bloch to study the electron behavior in crystalline solids. His method has been adapted to study the elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In a previous article (Farzbod and Leamy, 2009, “The Treatment of Forces in Bloch Analysis,” J. Sound Vib., 325(3), pp. 545–551), we clarified the treatment of internal forces. In this article, we borrow the insight from the previous work to detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. For example, we conclusively show that translational invariance is not a proper justification for invoking the existence of a “propagation constant,” and that in nonlinear media, this results in a flawed analysis. We also provide a simple, two-dimensional example, illustrating what the role stiffness symmetry has on the search for a band gap behavior along the edges of the irreducible Brillouin zone. This complements other treatments that have recently appeared addressing the same issue.

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.

scholarly journals Analysis of Vibration and Wave Propagation in Cylindrical Grid-Like Structures

2004 ◽  
Vol 11 (3-4) ◽  
pp. 311-331 ◽  
Author(s):  
Sang Min Jeong ◽  
Massimo Ruzzene
Keyword(s):  
Wave Propagation ◽  
Wave Attenuation ◽  
Periodic Structures ◽  
Interpolation Method ◽  
Dynamic Properties ◽  
Element Formulation ◽  
Two Dimensional ◽  
Combined Application ◽  
Cylindrical Grid ◽  
Phase Constant

The wave propagation in and the vibration of cylindrical grid structures are analyzed. The grids are composed of a sequence of identical elementary cells repeating along the axial and the circumferential direction to form a two-dimensional periodic structure. Two-dimensional periodic structures are characterized by wave propagation patterns that are strongly frequency dependent and highly directional. Their wave propagation characteristics are determined through the analysis of the dynamic properties of the unit cell. Each cell here is modelled as an assembly of curved beam elements, formulated according to a mixed interpolation method. The combined application of this Finite Element formulation and the theory of two-dimensional periodic structures is used to generate the phase constant surfaces, which define, for the considered cell lay-out, the directions of wave propagation at assigned frequencies. In particular, the directions and frequencies corresponding to wave attenuation are evaluated for cells of different size and geometry, in order to identify topologies with attractive wave attenuation and vibration confinement characteristics. The predictions from the analysis of the phase constant surfaces are verified by estimating the forced harmonic response of complete cylindrical grids, obtained through the assembly of the unit cells. The considered analysis provides invaluable guidelines for the investigation of the dynamic properties and for the design of grid stiffened cylindrical shells with unique vibration confinement characteristics.

Analysis of Vibration and Wave Propagation in Cylindrical Grid-Like Structures

2003 ◽  
Author(s):  
Sang Min Jeong ◽  
Massimo Ruzzene
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Periodic Structures ◽  
Dynamic Properties ◽  
Unit Cell ◽  
Element Formulation ◽  
Two Dimensional ◽  
Curved Beams ◽  
Cylindrical Grid ◽  
Phase Constant

The wave propagation in and the vibration of cylindrical grid structures are analyzed. The considered grids are composed of a sequence of identical elementary cells repeating along the axial and circumferential directions to form a two-dimensional (2D) periodic structure. Two-dimensional periodic structures are characterized by wave propagation patterns that are strongly frequency dependent and highly directional. Such unique characteristics can be utilized to design structures able to confine external perturbations to specified regions. The wave propagation characteristics of 2D periodic structures are determined through the analysis of the dynamic properties of the unit cell, which is described by its Finite Element mass and stiffness matrices. The cell is composed of curved beams to form a cylindrical grid. The combined application of the Finite Element formulation and the theory of 2D periodic structures yields the phase constant surfaces, which define, for the considered cell lay-out, the directions of wave propagation for assigned frequency values. The predictions from the phase constant surfaces analysis are verified by estimating the forced harmonic response of the complete grid. The results demonstrate the unique characteristics of this class of grid structures, and suggest how they may be designed to enhance attenuation capabilities of shell structures commonly used in aerospace or naval applications. Design configurations can be identified so that the transmission of vibrations towards specified locations and at certain frequencies is minimized. The study can be extended to include the optimization of the geometry and topology of the unit cell to achieve desired transmissibility levels in specified directions and for given excitation frequencies.

Acoustic wave propagation in two dimensional sheared ducted flows

1997 ◽  
Author(s):  
E. Longatte ◽  
P. Lafon ◽  
S. Candel ◽  
E. Longatte ◽  
P. Lafon ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Acoustic Wave ◽  
Two Dimensional ◽  
Acoustic Wave Propagation

Influence of the applied pressure of the transducer on the propagation speed of the ultrasonic wave in wood

Holzforschung ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar V.M. Carrasco ◽  
Rejane C. Alves ◽  
Mônica A. Smits ◽  
Vinnicius D. Pizzol ◽  
Ana Lucia C. Oliveira ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Applied Pressure ◽  
Pressure Gauge ◽  
Eucalyptus Grandis ◽  
Propagation Speed ◽  
Ultrasonic Waves ◽  
Ultrasonic Speed ◽  
Wave Propagation Speed ◽  
Propagation Technique ◽  
Non Destructive

Abstract The non-destructive wave propagation technique is used to estimate the wood’s modulus of elasticity. The propagation speed of ultrasonic waves is influenced by some factors, among them: the type of transducer used in the test, the form of coupling and the sensitivity of the transducers. The objective of the study was to evaluate the influence of the contact pressure of the transducers on the ultrasonic speed. Ninety-eight tests were carried out on specimens of the species Eucalyptus grandis, with dimensions of 120 × 120 × 50 mm. The calibration of the pressure exerted by the transducer was controlled by a pressure gauge using a previously calibrated load cell. The robust statistical analysis allowed to validate the experimental results and to obtain consistent conclusions. The results showed that the wave propagation speed is not influenced by the pressure exerted by the transducer.

scholarly journals Study on X-ray Induced Two-Dimensional Thermal Shock Waves in Carbon/Phenolic

Materials ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3553
Author(s):  
Dengwang Wang ◽  
Yong Gao ◽  
Sheng Wang ◽  
Jie Wang ◽  
Haipeng Li
Keyword(s):  
Numerical Simulation ◽  
Shock Wave ◽  
Wave Propagation ◽  
Thermal Shock ◽  
Anisotropic Material ◽  
Energy Deposition ◽  
X Rays ◽  
Two Dimensional ◽  
X Ray ◽  
Deposition Depth

Carbon/Phenolic (C/P), a typical anisotropic material, is an important component of aerospace and often used to protect the thermodynamic effects of strong X-ray radiation. In this paper, we establish the anisotropic elastic-plastic constitutive model, which is embedded in the in-house code “RAMA” to simulate a two-dimensional thermal shock wave induced by X-ray. Then, we compare the numerical simulation results with the thermal shock wave stress generated by the same strong current electron beam via experiment to verify the correctness of the numerical simulation. Subsequently, we discuss and analyze the rules of thermal shock wave propagation in C/P material by further numerical simulation. The results reveal that the thermal shock wave represents different shapes and mechanisms by the radiation of 1 keV and 3 keV X-rays. The vaporization recoil phenomenon appears as a compression wave under 1 keV X-ray irradiation, and X-ray penetration is caused by thermal deformation under 3 keV X-ray irradiation. The thermal shock wave propagation exhibits two-dimensional characteristics, the energy deposition of 1 keV and 3 keV both decays exponentially, the energy deposition of 1 keV-peak soft X-ray is high, and the deposition depth is shallow, while the energy deposition of 3 keV-peak hard X-ray is low, and the deposition depth is deep. RAMA can successfully realize two-dimensional orthotropic elastoplastic constitutive relation, the corresponding program was designed and checked, and the calculation results for inspection are consistent with the theory. This study has great significance in the evaluation of anisotropic material protection under the radiation of intense X-rays.

Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals

2000 ◽  
Vol 62 (4) ◽  
pp. 5711-5720 ◽  
Author(s):  
A. A. Asatryan ◽  
P. A. Robinson ◽  
L. C. Botten ◽  
R. C. McPhedran ◽  
N. A. Nicorovici ◽  
...  
Keyword(s):  
Refractive Index ◽  
Wave Propagation ◽  
Photonic Crystals ◽  
Two Dimensional
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