Multicell Homogenization of One-Dimensional Periodic Structures

2010 ◽  
Vol 132 (1) ◽  
Author(s):  
Stefano Gonella ◽  
Massimo Ruzzene
Keyword(s):  
Dynamic Behavior ◽  
Periodic Structures ◽  
Low Frequency ◽  
Structural Response ◽  
Fourier Series Expansion ◽  
Low Frequencies ◽  
One Dimensional ◽  
Physical Constraints ◽  
Unit Cells ◽  
Taylor Series Expansions

Much attention has been recently devoted to the application of homogenization methods for the prediction of the dynamic behavior of periodic domains. One of the most popular techniques employs the Fourier transform in space in conjunction with Taylor series expansions to approximate the behavior of structures in the low frequency/long wavelength regime. The technique is quite effective, but suffers from two major drawbacks. First, the order of the Taylor expansion, and the corresponding frequency range of approximation, is limited by the resulting order of the continuum equations and by the number of boundary conditions, which may be imposed in accordance with the physical constraints on the system. Second, the approximation at low frequencies does not allow capturing bandgap characteristics of the periodic domain. An attempt at overcoming the latter can be made by applying the Fourier series expansion to a macrocell spanning two (or more) irreducible unit cells of the periodic medium. This multicell approach allows the simultaneous approximation of low frequency and high frequency dynamic behavior and provides the capability of analyzing the structural response in the vicinity of the lowest bandgap. The method is illustrated through examples on simple one-dimensional structures to demonstrate its effectiveness and its potentials for application to complex one-dimensional and two-dimensional configurations.

scholarly journals Longitudinal wave propagation in a one-dimensional quasi-periodic waveguide

2019 ◽  
Vol 475 (2231) ◽  
pp. 20190392
Author(s):  
Vladislav S. Sorokin
Keyword(s):  
Wave Propagation ◽  
Wave Attenuation ◽  
Periodic Structures ◽  
Low Frequency ◽  
Periodic Waveguide ◽  
One Dimensional ◽  
Vibration Attenuation ◽  
Direct Separation ◽  
Longitudinal Wave Propagation ◽  
Spatial Modulations

The paper deals with the analysis of wave propagation in a general one-dimensional (1D) non-uniform waveguide featuring multiple modulations of parameters with different, arbitrarily related, spatial periods. The considered quasi-periodic waveguide, in particular, can be viewed as a model of pure periodic structures with imperfections. Effects of such imperfections on the waveguide frequency bandgaps are revealed and described by means of the method of varying amplitudes and the method of direct separation of motions. It is shown that imperfections cannot considerably degrade wave attenuation properties of 1D periodic structures, e.g. reduce widths of their frequency bandgaps. Attenuation levels and frequency bandgaps featured by the quasi-periodic waveguide are studied without imposing any restrictions on the periods of the modulations, e.g. for their ratio to be rational. For the waveguide featuring relatively small modulations with periods that are not close to each other, each of the frequency bandgaps, to the leading order of smallness, is controlled only by one of the modulations. It is shown that introducing additional spatial modulations to a pure periodic structure can enhance its wave attenuation properties, e.g. a relatively low-frequency bandgap can be induced providing vibration attenuation in frequency ranges where damping is less effective.

Propagation of Low-Frequency Elastic Disturbances in a Composite Material

1974 ◽  
Vol 41 (1) ◽  
pp. 97-100 ◽  
Author(s):  
W. Kohn
Keyword(s):  
Composite Material ◽  
Wave Equation ◽  
Dispersion Relation ◽  
Displacement Field ◽  
Low Frequency ◽  
Local Strain ◽  
Airy Functions ◽  
Low Frequencies ◽  
One Dimensional ◽  
Long Wavelength

In the limit of low frequencies the displacement u(x, t) in a one-dimensional composite can be written in the form of an operator acting on a slowly varying envelope function, U(x, t): u(x, t) = [1 + v1(x)∂/∂x + …] U(x, t). U(x, t) itself describes the overall long wavelength displacement field. It satisfies a wave equation with constant, i.e., x-independent, coefficients, obtainable from the dispersion relation ω = ω(k) of the lowest band of eigenmodes: (∂2/∂t2 − c¯2∂2/∂x2 − β∂4/∂x4 + …) U(x, t) = 0. Information about the local strain, on the microscale of the composite laminae, is contained in the function v1(x), explicitly expressible in terms of the periodic stiffness function, η(x), of the composite. Appropriate Green’s functions are constructed in terms of Airy functions. Among applications of this method is the structure of the so-called head of a propagating pulse.

Propagation of Low Frequency Elastic Disturbances in a Three-Dimensional Composite Material

1975 ◽  
Vol 42 (1) ◽  
pp. 159-164 ◽  
Author(s):  
W. Kohn
Keyword(s):  
Secular Equation ◽  
Displacement Vector ◽  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Low Frequencies ◽  
One Dimensional ◽  
Coupled Partial Differential Equations ◽  
Constant Coefficients ◽  
The Mean

This paper is a generalization to three dimensions of an earlier study for one-dimensional composites. We show here that in the limit of low frequencies the displacement vector ui(r,t) can be written in the form ui (r,t) = (∂ij + vijl (r) ∂/∂xl + …) Uj (r,t). Here Uj (r,t) is a slowly varying vector function of r and t which describes the mean displacement of each cell of the composite. Its components satisfy a set of three coupled partial differential equations with constant coefficients. These coefficients are obtainable from the three-by-three secular equation which yields the low-lying normal mode frequencies, ω(k). Information about local strains is contained in the function vijl(r), which is characteristic of static deformations, and is discussed in detail. Among applications of this method is the structure of the head of a pulse propagating in an arbitrary direction.

Multi-Cell Homogenization of Elastic Periodic Domains

2006 ◽  
Author(s):  
Stefano Gonella ◽  
Massimo Ruzzene
Keyword(s):  
Band Gap ◽  
Dynamic Behavior ◽  
Series Expansion ◽  
Taylor Series ◽  
Dynamic Properties ◽  
Fourier Series Expansion ◽  
Periodic Domain ◽  
Low Frequencies ◽  
Gap Characteristics ◽  
Periodic Domains

Recently, much attention has been devoted to the application of homogenization methods for the prediction of the dynamic behavior of periodic domains. One of the most popular techniques consists in the application of the Fourier Transform in space which allows the application of Taylor series approximations for low frequencies/long wavelengths. This method provides continuum equations which approximate the dynamic behavior of the considered periodic domain over a range of frequencies which is defined by the order of the considered Taylor series expansion. This technique is very effective, but suffers from two major drawbacks. First, the order of the Taylor expansion, and therefore the frequency range of approximation, is limited by the corresponding order of the continuum equations and by the number of boundary conditions which may be imposed in accordance with the physical constraints on the system. Second, the approximation at low frequencies does not allow capturing the band gap characteristics of the periodic domain. An attempt at overcoming the latter can be made by applying the Fourier series expansion to a macro cell spanning two (or more) irreducible unit cells of the periodic medium. This multi-cell approach allows the description of both average and intra-cell behavior of the domain, and approximates dispersion relations and corresponding dynamic properties at low frequencies and at frequencies close to the lower band gap. The resulting continuum equations are therefore capable of reproducing in part the band-gap characteristics of the structure. The proposed methodology is tested on simple one-dimensional and two-dimensional structures, which illustrate the method and show its effectiveness.

Analysis of an Ultra-Low Frequency and Ultra-Broadband Phononic Crystals Silencer with Small Size

2019 ◽  
Vol 27 (02) ◽  
pp. 1850026 ◽  
Author(s):  
Jiangwei Liu ◽  
Dianlong Yu ◽  
Jihong Wen ◽  
Zhenfang Zhang
Keyword(s):  
Wave Propagation ◽  
Periodic Structures ◽  
Low Frequency ◽  
Phononic Crystals ◽  
Frequency Noise ◽  
Duct System ◽  
Frequency Range ◽  
Expansion Chamber ◽  
Band Formation ◽  
Low Frequencies

Existing research shows that acoustic BG in a certain frequency range can be realized by installing an expansion chamber on duct system, but the problems of broadband and size limitations at low frequencies remain to be researched. The study of acoustic and elastic wave propagation in artificial periodic structures has received increasing attention for many decades, and the presence of bandgap (BG) in phononic crystals (PCs), which inhibits elastic/acoustic wave propagation within the BG’ frequency range, supplies a new way to control noise and vibrations in duct system. Based on PC theory, a duct silencer backed with a gas–liquid expansion chamber is proposed to enhance the acoustic performance of low-frequency noise attenuation. The transfer matrix method (TMM) is used to investigate the acoustic BG properties. The influences on the BG properties of some key parameters are analyzed, and the band formation mechanism is revealed by the law of energy conservation. The results show that silencers with a small size can effectively attenuate ultra-low frequencies and ultra-broad bands.

Vibration isolation characteristics of finite periodic tetra-chiral lattice coating filled with internal resonators

2016 ◽  
Vol 230 (16) ◽  
pp. 2840-2850 ◽  
Author(s):  
Dawei Zhu ◽  
Xiuchang Huang ◽  
Hongxing Hua ◽  
Hui Zheng
Keyword(s):  
Band Gap ◽  
Vibration Isolation ◽  
Periodic Structures ◽  
Low Frequency ◽  
Band Gaps ◽  
Stiffened Plate ◽  
Frequency Range ◽  
Vibration Attenuation ◽  
Unit Cells ◽  
Internal Resonators

Owing to their locally resonant mechanism, internal resonators are usually used to provide band gaps in low-frequency region for many types of periodic structures. In this study, internal resonators are used to improve the vibration attenuation ability of finite periodic tetra-chiral coating, enabling high reduction of the radiated sound power by a vibrating stiffened plate. Based on the Bloch theorem and finite element method, the band gap characteristics of tetra-chiral unit cells filled with and without internal resonators are analysed and compared to reveal the relationship between band gaps and vibration modes of such tetra-chiral unit cells. The rotational vibration of internal resonators can effectively strengthen the vibration attenuation ability of tetra-chiral lattice and extend the effective frequency range of vibration attenuation. Two tetra-chiral lattices with and without internal resonators are respectively designed and their vibration transmissibilities are measured using the hammering method. The experimental results confirm the vibration isolation effect of the internal resonators on the finite periodic tetra-chiral lattice. The tetra-chiral lattice as an acoustic coating is applied to a stiffened plate, and analysis results indicate that the internal resonators can obviously enhance the vibration attenuation ability of tetra-chiral lattice coating in the frequency range of the band gap corresponding to the rotating vibration mode of internal resonators. When the soft rubber with the internal resonators in tetra-chiral layers has gradient elastic modulus, the vibration attenuation ability and noise reduction of the tetra-chiral lattice coating are basically enhanced in the frequency range of the corresponding band gaps of tetra-chiral unit cells.

Plate-type metamaterials for extremely broadband low-frequency sound insulation

2018 ◽  
Vol 32 (03) ◽  
pp. 1850019 ◽  
Author(s):  
Xiaopeng Wang ◽  
Xinwei Guo ◽  
Tianning Chen ◽  
Ge Yao
Keyword(s):  
Transmission Loss ◽  
Structural Parameters ◽  
Low Frequency ◽  
Thin Plates ◽  
Sound Insulation ◽  
Frequency Noise ◽  
Low Frequencies ◽  
Acoustic Metamaterial ◽  
Unit Cells ◽  
Plate Type

A novel plate-type acoustic metamaterial with a high sound transmission loss (STL) in the low-frequency range ([Formula: see text]1000 Hz) is designed, theoretically proven and then experimentally verified. The thin plates with large modulus used in this paper mean that we do not need to apply tension to the plates, which is more applicable to practical engineering, the achievement of noise reduction is better and the installation of plates is more user-friendly than that of the membranes. The effects of different structural parameters of the plates on the sound-proofed performance at low-frequencies were also investigated by experiment and finite element method (FEM). The results showed that the STL can be modulated effectively and predictably using vibration theory by changing the structural parameters, such as the radius and thickness of the plate. Furthermore, using unit cells of different geometric sizes which are responsible for different frequency regions, the stacked panels with thickness [Formula: see text]16 mm and weight [Formula: see text]5 kg/m2 showed high STL below 2000 Hz. The acoustic metamaterial proposed in this study could provide a potential application in the low-frequency noise insulation.

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.

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