Formation Of Stop Band Research Articles

Sound waves in continuum models of periodic sonic black holes

In this work we address the study of continuum models of periodic sonic black holes (SBHs) at duct terminations. A conventional SBH combines a decreasing power-law profile with a wall admittance in such a way that sound waves entering the SBH slow down, while their amplitude increases and their wavelength decreases as they approach the duct termination. In an ideal scenario, the wave is trapped and dissipated inside the SBH so that no reflection occurs. Instead of dealing with conventional SBHs, we are interested here in periodic SBHs. These have the potential to extend the application range of SBHs below the cut-on frequency thanks to the formation of stopbands, among other significant aspects. First, the case of an SBH cell placed between semi-infinite ducts is analyzed and it is shown how it reduces not only sound reflection but also sound transmission. The acoustic pressure inside the SBH is governed by a modified Webster equation that is solved in weak form using a basis of Gaussian functions. The dependence of the reflection, transmission and absorption coefficients for different SBH parameters and boundary conditions is analyzed in detail. Next, the case of an infinite periodic SBH lattice is considered. The nullspace method (NSM) is used to impose the periodic boundary conditions of the problem. Complex dispersion curves are computed and the separate roles played by the SBH power-law profile and wall admittance on the dispersion curves and pressure distribution inside the cells are examined. Bragg scattering turns out to be the main mechanism behind bandgap formation and wall admittance is essential to achieve the SBH effect. A parametric study follows, showing the influence of the damping, the residual radius, the SBH order and its length on the imaginary part of the complex dispersion curves. Finally, the reflection, transmission and absorption coefficients for a finite 3-cell SBH are investigated. Although most of the research on SBHs has focused on practical application designs, it is believed that a thorough understanding of the performance of continuum models of SBHs could be very useful for the former.

Modal-Based Analysis for Aiding 3D Elastic Metastructure Design

The engineered topological structures of the unit cell endow elastic metamaterials (EMMs) with the extraordinary capability to attenuate elastic waves. In real-life scenarios, a practical EMM (i.e., metastructure), consisting of a limited number of unit cells, is the truncation of the infinite EMM, which may detriment seriously the attenuation capability. To understand the mechanism behind the detriment, an efficient way to evaluate and analyze the stopbands for metastructures is essential. Instead of relying on the commonly adopted frequency response analysis to characterize stopbands for metastructures, which are sensitive to frequency sweeping steps and short of information of truncated resonance affecting wave attenuation performance, a novel modal-based method (MM) is proposed to assess the wave attenuation of 3D metastructures. Specifically, the modal-based analysis scheme incorporating the modal superposition principle and a freshly developed layer-based strain energy ratio (SER) measurement method is proposed. Through employing the MM, the opening and ceasing of the stopbands are evaluated by characterizing dominant eigenmodes developed in metastructures. Accordingly, the influences of eigenmodes induced by different mechanisms on wave attenuation performance are investigated, and the stopband formation mechanisms are elaborated. Furthermore, under the guidance of a freshly proposed modal-based design framework, metastructures with modified geometrical parameter settings (GPSs) are demonstrated with enlarged normalized bandwidths (NBs) at lower frequencies, which show improved applicability in multiple engineering disciplines.

On the formation of a super stop-band in finite mono-coupled periodic structures using an array of vibration absorbers: Controlling parameters and physical insight

There are two fundamental strategies in the design of metamaterials used for noise and vibration control. The first uses structural periodicity and explores the Bragg scattering to produce frequency ranges where waves do not propagate or are severely attenuated (stop-bands). This condition is usually limited by the constitutive size of the structural elements and tends to occur at higher frequencies. A second strategy consists of adding vibration absorbers to produce a resonance(s), and this can occur at much lower frequencies. The addition of periodically attached vibration absorbers alone creates a periodic medium, and Bragg stop-bands also occurs in this situation. There is a particular condition where the Bragg and local resonance stop-bands can be combined to produce an ultra-wide or a super stop-band. This paper investigates the conditions required to produce a super stop-band in a finite mono-coupled periodic system composed of symmetric cells, each with a vibration absorber attached. Using a lumped parameter system, the dynamic features and the transmissibility of a single cell are determined, and the physical mechanisms behind the super stop-band formation are investigated, including the absorber tuning frequency and the influence of damping. Two experimental setups to support the theoretical findings are described. One involves a lumped-parameter system and the other involves a rod, both with a vibration absorber attached. The results show that a smooth super stop-band only occurs when the vibration absorber is tuned to the free-free natural frequency of the single host cell, creating a union between the Bragg stop-bands due to mass and stiffness of the absorber, and that the damping in the absorber should be equal to the damping in the host cell.

Broadband noise mitigation using coupled Helmholtz resonators: a numerical study

In this work we investigate a periodic structure in the frequency range from 20 Hz to 5500 Hz designed for broadband noise insulation. The considered unit cell consists of a simple structure: a pair of polymer pipes with slits carved along the axes, representing two coupled Helmholtz resonators. In order to develop a design with a broad band gap, we analyze the eigenmodes of the infinite two-dimensional structure considering their symmetry and interaction. This analysis is supported by parametric optimization of the resonator geometry. The obtained band diagram is compared with numerically determined transmission coefficient of a finite structure based on the same unit cell. The number of unit cells of the finite structure is chosen to be sufficient for demonstration of insulating properties and stop band formation. Furthermore, we analyze how the transmission coefficient is linked to the pressure field distribution inside the resonators. Owing to the simplicity of the geometry, the obtained results may become a basis for development of budget-friendly passive systems for broadband noise insulation within the audible range of frequencies.

Design of elastic metamaterials with ultra-wide low-frequency stopbands via quantitative local resonance analysis

Elastic metamaterials (EMMs) with engineered bandgaps have numerous applications in medical, transport, and aerospace domains due to their extraordinary performance in manipulating elastic waves. Nevertheless, developing an efficient method to guide designs of EMMs with low-frequency and wide frequency stopband is still a challenging problem in this field. In this paper, a quantitative local resonance analysis scheme is proposed for efficient design of three-dimensional (3D) latticed EMMs with broad and complete wave attenuation band at low frequency. To acquire a low starting frequency, a diatomic mass spring analytic model is applied to guide the modification of structure and adjustment of geometrical parameters of the EMM, and thus its effective stiffness and mass values affecting the starting frequency are altered desirably. With the aim of enlarging stopband, a strategy to analyze stopband formation mechanism is proposed by virtue of modal superposition principle. Dominant vibration modes developed on the EMM are found with modal superposition principle and these modes are further characterized with the aid of a quantitative local resonance measurement method aiming at exploring vibration features on the EMM. Upon the revealing of wave attenuation mechanism, we show the low-frequency stopband for the EMM can be widened via altering vibration characteristics of dominant modes that are responsible for stopband closing. The proposed new analysis scheme as well as the EMM structure facilitate the development of structure designs for elastic wave attenuation.

Two-path phonon interference resonance induces a stop band in a silicon crystal matrix with a multilayer array of embedded nanoparticles

In this work, we report a new stop-band formation mechanism by performing the atomistic Green's function calculation and the wave-packet molecular dynamics simulation for a system with germanium-nanoparticle array embedded in a crystalline silicon matrix. When only a single nanoparticle is embedded, the local resonance, induced through destructive interference between two different phonon wave paths, gives rise to several sharp and significant transmittance dips. On the other hand, when the number of embedded nanoparticles further increases to ten, a stop band with complete phonon reflection is formed due to the two-path resonance Bragg-like phonon interference. The wave packet simulations further uncover that the stop band originates from the collective phonon resonances at the embedded nanoparticles. Compared with the traditional stop-band formation mechanism that is the single-path Bragg reflection, the resonance mechanism has a significant advantage in not requiring the strict periodicity in the embedded nanoparticles array. We also demonstrate that the stop band can significantly suppress thermal conductance in the low-frequency regime. Our work provides a robust, scalable. and easily modulable stop-band formation mechanism. which opens a new degree of freedom for phononics-related heat control.

Bandwidth‐controllable band‐stop filter using spoof surface plasmon polaritons

This article presents a bandwidth-controllable band-stop filter based on spoof surface plasmon polaritons (SSPPs) transmission line in the microwave frequency band. The approach of introducing rectangular slots into the corrugation strips offers a greater degree of flexibility in the SSPPs unit cell design. Meanwhile, the high-order mode can be pulled to the low frequency segment of interest. The lower and upper edge of the stop band of this filter, therefore, can be controlled independently and conveniently. An equivalent circuit model of this unique unit cell is proposed for analyzing stop band formation and guiding the design of this SSSPs filter. Both numerical simulations and measured results demonstrate excellent stop band performance of the filter. The proposed filter may have potential applications in the SSPPs-based communication system.

Flexural wave scattering by varying-thickness annular inclusions on infinite thin plates

This paper proposes a semi-analytical method for flexural wave scattering by cylindrical annular inclusions of varying thickness on an infinite thin plate. The thickness of inclusions is assumed to vary in arbitrary power form along the radial direction. In the numerical method, analytically linear relationships between expansion coefficients of exciting and scattered fields around each inclusion are firstly derived by means of closed form solutions of a hypergeometric governing equation of varying-thickness thin plates, and boundary conditions at the interface between different media. Then, linear algebraic equations containing only expansion coefficients of exciting fields are set up with the aid of foregoing linear relations. Expansion coefficients of exciting fields are finally solved by a numerical collocation method. Numerical examples of single and multiple scattering are designed to demonstrate the applicability of the method. The results of single scattering show the change of thickness variations allows tuning local resonances of inclusions to specified frequencies. Furthermore, displacement amplitude spectra of flexural waves for multiple scattering have captured the details of stop band formation of square and hexagonal arrangements of varying-thickness inclusions. It is also demonstrated that stop bands of flexural wave propagation can be controlled by changing stiffness gradients of varying-thickness inclusions.

Realisation of a locally resonant metamaterial on the automobile panel structure to reduce noise radiation

Abstract This study aims to demonstrate the reduction in noise and vibration of an automobile dash panel structure by applying a locally resonant metamaterial (LRM). LRMs are implemented by adding a subwavelength array of local resonators to a host structure. Here, the minimum repeated structure is called a unit cell. Owing to dynamic absorption in a unit cell, LRM has a stop band (or also called a band gap), which is a frequency band in which waves cannot propagate. Taking advantage of the stop band, various studies have been conducted on mechanical noise and vibration problems. However, no case has yet been published regarding the application of LRMs to industrial structures such as automobile dash panels because of their complex shape. To overcome the difficulties of LRM realisation in dash panels, an attachable local resonator (ALR) is proposed in this study. The proposed ALR consists of a local resonator and a permanent magnet to be attached on the dash panel surface. To predict the stop band formation, a finite element-based unit cell analysis method is developed, and several ALRs are designed based on the stop band. The designed ALRs are applied to an actual automobile dash panel structure and the acoustic and vibration responses are measured. The measurement results demonstrate that the vibration and noise radiation from the dash panel structure are greatly reduced in the designed stop bands.

Multiple scattering and stop band characteristics of flexural waves on a thin plate with circular holes

A numerical procedure is proposed for the multiple scattering analysis of flexural waves on a thin plate with circular holes based on the Kirchhoff plate theory. The numerical procedure utilizes the wave function expansion of the exciting as well as scattered fields, and the boundary conditions at the periphery of holes are incorporated as the relations between the expansion coefficients of exciting and scattered fields. A set of linear algebraic equations with respect to the wave expansion coefficients of the exciting field alone is established by the numerical collocation method. To demonstrate the applicability of the procedure, the stop band characteristics of flexural waves are analyzed for different arrangements and concentrations of circular holes on a steel plate. The energy transmission spectra of flexural waves are shown to capture the detailed features of the stop band formation of regular and random arrangements of holes. The increase of the concentration of holes is found to shift the dips of the energy transmission spectra toward higher frequencies as well as deepen them. The hexagonal hole arrangement can form a much broader stop band than the square hole arrangement for flexural wave transmission. It is also demonstrated that random arrangements of holes make the transmission spectrum more complicated.

Phononic Crystal Artifacts for Real-Time In Situ Quality Monitoring in Additive Manufacturing

Additive manufacturing (AM) is rapidly becoming a local manufacturing modality in fabricating complex, custom-designed parts, providing an unprecedented form-free flexibility for custom products. However, significant variability in part geometric quality and mechanical strength due to the shortcomings of AM processes has often been reported. Presently, AM generally lacks in situ quality inspection capability, which seriously hampers the realization of its full potential in delivering qualified practical parts. Here, we present a monitoring approach and a periodic structure design for developing test artifacts for in situ real-time monitoring of the material and bonding properties of a part at fiber/bond-scale. While the production method used in current work is filament based, the proposed approach is generic as defects are always due to materials in a bonding zone and their local bonding attributes in any production modality. The artifact design detailed here is based on ultrasonic wave propagation in phononic coupons consisting of repeating substructures to monitor and eventually to assess the bond quality and placement uniformity—not only for geometry but also for defect states. Periodicity in a structure leads to the dispersion of waves, which is sensitive to geometric/materials properties and irregularities. In this proof-of-concept study, an experimental setup and basic artifact designs are described and off-line/real-time monitoring data are presented. As a model problem, the effects of printing speed on the formation of stop bands, wave propagation speeds and fiber placement accuracy in samples are detected and reported.

Periodicity effects of axial waves in elastic compound rods

Floquet analysis is applied to the Bernoulli–Euler model for axial waves in a periodic rod. Explicit asymptotic formulae for the stop band borders are given and the topology of the stop band pattern is explained. Eigenfrequencies of the symmetric unit cell are determined by the Phase-closure Principle, and their correspondence with stop band formation is shown. Steady-state and transient dynamics of a periodic rod of finite length are analysed numerically and the difference in structural response when excitation is done in either stop- or pass bands is demonstrated. A physical interpretation of the underlying mechanisms of stop bands is proposed.

Multiple Scattering of Elastic Waves in Unidirectional Composites with Coated Fibers

A computational method for analyzing the two-dimensional multiple scattering of elastic waves in unidirectional fiber-reinforced media, consisting of elastic cylindrical fibers arranged in an infinitely extended elastic matrix, is presented. The method is based on the eigenfunction expansion of the displacement potentials and the numerical collocation technique to obtain the expansion coefficients. In the present study, the formulation is extended to incorporate the presence of a coating layer between the fibers and the matrix. As an example, the transmission characteristics of longitudinal as well as transverse waves in a unidirectional composite with square array of fibers are analyzed, and the phase velocities in the composite are evaluated. The effect of the coating layer on the wave transmission and stop-band formation behavior is also demonstrated.

Computational multiple scattering analysis of elastic waves in unidirectional composites

A numerical procedure is presented in this paper for the two-dimensional, time-harmonic elastodynamic multiple scattering problems for unidirectional fiber-reinforced composites. The proposed procedure is based on the eigenfunction expansion of the displacement potentials and the numerical collocation method to solve the expansion coefficients, and is capable of modeling arbitrary fiber arrangements. To demonstrate the applicability of the procedure, the P and SV wave propagation characteristics in unidirectional fiber-reinforced composites are analyzed for different fiber arrangements and fiber volume fractions. The simulated results are shown to capture the detailed features of the local wave fields in the composites accompanying the mode conversion. From the computed wave fields, the effective phase velocities of the composites are identified as functions of the frequency, and found to be in good agreement with the predictions of a micromechanical model for random composites. The energy transmission spectra of the P and SV waves are also demonstrated, which exhibit the stop-band formation for the composites with regular fiber arrangements.

Random lasing in the composites consisting of photonic crystals and semiconductor nanowires

A well designed material composed of photonic crystals (PCs) and semiconductor nanowires was proposed. To illustrate our working principle, the nanocomposites consisting of SnO2 nanowires and PCs based on Tb(OH)3/SiO2 core/shell nanospheres have been synthesized and characterized. It is found that lasing behavior can be easily achieved using this composite material. The light confined inside the PCs due to the formation of stop band can be extracted along SnO2 nanowires. The observation of random lasing behavior indicates that the composites developed here open a route for the creation of optoelectronic devices.

Plasmonic stop band formation in a metal–insulator–metal ring with a narrowgap

Using finite-difference time-domain simulations it is shown that a metal–insulator–metal(MIM) ring with a narrow gap exhibits plasmonic stop bands (PLSBs) in a similar waythat a microstrip ring with a narrow gap exhibits photonic stop bands. A narrow gapcreated in the middle of the upper half of a regular MIM ring causes plasmonic fieldsto experience multiple reflections at the discontinuity introduced by the gap,which results in the formation of PLSBs. The PLSBs appear as transmissiondips around the resonant frequencies of the isolated MIM ring with the narrowgap. The centre frequency and the range of PLSBs are shown to be controllableby varying the width of the narrow gap. The proposed design for a MIM ringin which PLSBs exist is expected to reduce the size of the plasmonic devices.

Fabrication of Metallodielectric Photonic Crystals with a Diamond Structure and their Microwave Properties

Three‐dimensional (3D) metallodielectric photonic crystals with a diamond structure were fabricated in order to investigate the formation of stop bands and the absorption ability for microwaves with the dielectric absorbing media embedded into the 3D metal lattice. First, the metallic photonic crystals were prepared by filling the epoxy molds formed by stereolithography with a metal alloy having a low melting point of 70°C, followed by removal of the molds. The metallodielectric photonic crystals were then fabricated by infiltrating the porous metal crystal with a SiC/polyester mixture. The lattice constant of photonic crystals was 15 mm. The effects of different aspect ratios of diamond lattice rods, number of metallic lattice units along Γ‐L 〈1 1 1〉, Γ‐X 〈1 0 0〉, and Γ‐K 〈1 1 0〉 directions, and metallodielectric samples along the Γ‐X 〈1 0 0〉 direction on the formation of stop band and microwave absorption ability were investigated in the frequency range from 3 to 30 GHz. Metallodielectric photonic crystals formed showed good absorption ability. The measured transmission spectra of the metallic and metallodielectric crystals agreed well with the simulation of the transmission line modeling method.

Non-Bloch plasmonic stop-band in real-metal gratings

Recent studies of plasmon surface wave (PSW) propagation in short-period perfectly conducting gratings have shown formation of stop-band that are not linked to the interaction between two (counter) propagating surface waves. We study the properties of this stop-band in real metals. While for both perfectly conducting and real metals the propagation constant of PSW grows with the groove height, the stop-band in real metals appears for groove heights significantly smaller than in perfect metals. A physical explanation of the formation of the stop-band is proposed both by using a homogenisation of the corrugated layer and by analysis of the tangential electric field component.

NDE via stop band formation in fiber reinforced composites having square fiber arrangements

Examples of full scale simulations of wave propagation through fiber reinforced composites are presented and from these the formations of stop bands in the response spectra are observed. The variations in the cut-off and recovery frequencies of the stop bands due to changes in the microstructural parameters of the composite suggest a useful tool in NDE.

Impurity effects in coaxial-connector photonic crystals: A quasi-one-dimensional periodic system

A one-dimensional coaxial connector photonic crystal has constructed from a series of standard high frequency coaxial cable ``tee'' and ``barrel'' connectors, forming a quasi-one-dimensional periodic system with periodicities comparable to microwave wavelengths. At the appropriate frequencies, ``Bragg-like'' reflections of the gigahertz-frequency sinusoidal signals from the open ends of the tee connectors result in the formation of stop bands for which transmission of signals at these frequencies is forbidden. Changing the amplitude or/and phase of these reflections is analogous to the introduction of defects (impurities) in the periodic structure and is evidenced by appearance of impurity peaks of enhanced transmission in the stop bands. This system readily allows fixing of the defect location within a unit cell to one particular configuration. It also allows a direct control over the location of defects relative to each other. By using this system a wide variety of impurity-induced effects in photonic crystals such as surface effects, inter-impurity interactions, and stop band widening have been demonstrated.