Local Resonance Bandgaps Research Articles

Realization of topological Bragg and locally resonant interface states in one-dimensional metamaterial beam-resonator-foundation system

Abstract In this study, the one-dimensional (1D) metamaterial beam-foundation system is innovatively improved into a metamaterial beam-resonator-foundation system by inserting resonators into the elastic foundation for ultra-low frequency vibration attenuation and enhanced topological energy trapping. Abundant band gap characteristics are obtained including quasi-static band gap starting from 0 Hz, Bragg scattering band gaps (BSBGs), and local resonance band gaps (LRBGs). Five band folding points are obtained through the band folding mechanism which can be opened by tuning inner and outer resonance parameters. However, only three band folding induced band gaps support mode inversion and Zak phase transition, including one BSBG and two LRBGs. The topological inversion in LRBGs is rarely reported in the 1D mechanical system, which can induce topological locally resonant interface states. The underlying physical mechanism of the topological phase transition in LRBG is revealed, which results from the topological inversion band gap transition from an initial BSBG to a LRBG with resonance parameters changes. Different from conventional 1D topological metamaterials that merely utilize local resonance to lower the band frequency and achieve subwavelength topological states in BSBGs, the topological interface states in LRBGs can localize wave energy to fewer unit cells near the interface, exhibiting enhanced energy localization capacity. The topologically protected interface states are validated with defective cases, demonstrating the potential of topological metamaterials for robust energy harvesting. This study provides new insights into the topological theory of 1D mechanical systems and contributes to the development and implementation of multi-functional devices integrating vibration attenuation and energy trapping.

Low-frequency multiple topological interface modes in metamaterial beams with quasi-zero-stiffness resonators

To achieve multiple low-frequency topological interface modes (TIMs) while maintaining lightweight and load-bearing capacities in metamaterial beams, we propose novel topological beams equipped with double quasi-zero-stiffness (QZS) resonators. Following the introduction of the negative stiffness (NS) mechanism, we derived the dispersion relations of the topological beams with double QZS resonators using the transfer matrix method (TMM), and performed comparative and parametric analyses of stiffness ratio on folded Dirac cones (DCs) and locally resonant band gaps (LRBGs). Thanks to the NS mechanism and the double degrees of freedom (DOFs) in the QZS resonators, we can realize double low-frequency DCs (< 500 Hz) in metamaterial beams. It is notable that adjusting the stiffness of oblique springs can tune the resonators’ effective stiffness, implying that we can custom the frequency of DCs and LRBGs by adjusting the QZS resonator. By altering the distance between two QZS resonators, we obtain two topologically distinct unit cells. Transmission simulations confirm the presence of multiple low-frequency TIMs at the interface between these two domains. We also find that the effect of mass and geometric defects at various positions on TIMs is minimal. The proposed metamaterial beams could inspire innovative low-frequency vibration energy harvesters and lightweight topological devices.

A viscoelastic metamaterial beam for integrated vibration isolation and energy harvesting

Locally resonant metamaterials have low-frequency band gaps and the capability of converging vibratory energy in the band gaps at resonant cells. It has been demonstrated by several researchers that the dissipatioin of vibratory energy within the band gap can be improved by using viscoelastic materials. This paper designs an integrated viscoelastic metamaterial for energy harvesting and vibration isolation. The viscoelastic metamaterial is achieved by a viscoelastic beam periodically arrayed with spatial ball-pendulum nonlinear energy harvesters. The nonlinear resonator with an energy harvesting function is achieved by placing a free-rolling magnetic ball in a spherical cavity with an additional induction coil. The dynamic equations of viscoelastic metamaterials under transverse excitation are established, and the energy harvesting and vibration isolation characteristics within the dispersion relation of viscoelastic metamaterials are analyzed. The results show that the vibrations of the main body of the viscoelastic metamaterial beam are significantly suppressed in the frequency range of the local resonance band gap. At the same time, the elastic waves are limited in the nonlinear resonator with an energy harvesting function, which improves the energy output. Finally, an experimental platform of viscoelastic metamaterial vibration is established for validation purposes.

Research on targeted modulation of elastic wave bandgap in cantilever-structured piezoelectric Phononic crystals

Piezoelectric phononic crystals (PPCs) have the advantages of simplicity and efficiency in controlling elastic wave bandgaps, which can be utilized to solve NVH problems with varying frequency features, such as the low-frequency vibrations caused by the time-varying power source (i.e., motor) in the subframe and underbody of new energy vehicles. However, most of the previous studies on PPCs bandgap control are qualitative, lacking the knowledge of bandgap targeted modulation. This paper analyzes the locally resonant (LR) bandgap characteristics of PPCs with shunting circuits, and aims to achieve precise control of bandgap position by circuit parameters. A theoretical research model of cantilever-structured PPCs is established firstly, and its accuracy is verified through finite element analysis and experiments. Subsequently, the formation mechanism of the LR bandgap is analyzed, and a bandgap characteristic frequency estimation method is proposed, along with an analysis of the influence of electrical parameters using this method. Finally, from the perspective of bandgap practicality, the variable capacitance method is proposed, which can quickly calculate the capacitance value through an estimation formula under the arbitrary designation of target bandgap in the low-frequency range of 200 Hz∼500 Hz. The results show that the actual bandgap position is very close to the target bandgap position, and the bandgap always keeps enough bandwidth and attenuation amount. This method has broad prospects in the field of vibration attenuation where frequency characteristics are variable.

Low-frequency vibration and noise control in sandwiched composite locally resonant metamaterials-embedded plate structures

This study proposes a sandwiched composite locally resonant metamaterial (SLRM) system and SLRM-embedded plate structure (SLRMeP) to effectively control low-frequency vibrations and sound radiation. The wave control mechanism and configuration of the proposed system are more suitable and realistic to address practical low-frequency vibro-acoustic problems. A numerical model was proposed based on the material properties, unit dimensions, and mass ratios to determine the local resonance characteristics and bandgap formation. The experimental results on a full-scale SLRMeP measuring 3000 × 4200 × 210 mm confirmed the efficacy of the local resonance bandgap for controlling vibrations and sound radiation, achieving a 94.08% reduction in the acceleration response and a 15.13 dB reduction in the sound pressure level. Additionally, variations in mass ratio, achieved by altering the mass density or dimensions, yield distinct bandgap behaviors, offering strategies to enhance vibro-acoustic performance.

Hybrid behaviors of RC metaslab combining bandgap and isolation for broadband vibration control

In this study, a novel method is introduced, integrating the principles of the local resonance bandgap (LRBG) with vibration isolation techniques to achieve broadband flexural vibration control in plate structures. This methodology employs locally resonant metamaterial units incorporated into a reinforced concrete (RC) slab, making it suitable for real-world engineering applications. Concurrently, the construction specifics associated with this methodology are elucidated. The mechanisms behind LRBG and vibration isolation within the metaslab were examined using numerical analyses. From these analyses, the frequency range conducive for broadband flexural vibration control was estimated, factoring in both material properties and geometric dimensions. Experimental results on a full-scale RC metaslab, measuring 4,200 × 3,000 × 210 mm, validate that the fabricated metaslab introduces a broadband vibration control region attributed to both the LRBG and vibration isolation phenomena. These findings closely align with the numerical estimations. All bending modal responses inherent in the unmodified RC slab were encompassed within this broadband frequency range. Furthermore, the maximal and initial four modal responses of the RC metaslab were attenuated by up to 81.19 %, 99.57 %, 99.94 %, 99.71 %, and 99.79 %, respectively.

Low-frequency band gaps in quasi-zero stiffness locally resonant metamaterial shaft

The suppression of low-frequency torsional vibration has always been a difficult problem in the field of vibration control. Recently, metamaterials have attracted considerable attention due to their superior vibration insulation properties, at least in some targeted and tunable band gaps ranges. In particular, the position of the local resonance band gaps is related to the resonance frequency of the local resonator (LR). In this paper, a novel quasi-zero-stiffness torsional LR is proposed and a metamaterial shaft is devised by periodically attaching this LR onto a uniform shaft. The statics analysis for the torsional LR is conducted, and the propagation characteristics of the torsional vibration in the metamaterial shaft is studied with the transfer matrix theory and the Galerkin method. The analytical dispersion relation and the band gap structure indicate the existence of low frequency band gaps. And the location of the band gaps and the attenuation ability of torsional vibration can be effectively controlled by tuning the structural parameters of LR and the material of shaft, which provide a theoretical basis and guidance for addressing the challenge of low-frequency torsional vibration suppression.

An ultra-wideband plasmonic reflector based on local resonant bandgap and Bragg bandgap

Unlike previous reports that utilized periodic modulation of insulation layer thickness or dielectrics, we propose the use of split ring resonators (SRRs) and their arrays to modulate the propagation characteristics of MIM plasmon waveguides. Due to the strong resonance backscattering of SRR, resonance transmission valleys appear in the transmission spectra of MIM waveguides. Changing the size of SRR can achieve continuously adjustable positions of resonance transmission valleys. The introduction of SRRs periodic arrays will result in two bands (bandgaps) with transmission minimum in the transmission spectra. Combining the dependence of the two bandgaps on the array period and the field distribution, the two bandgaps are Bragg bandgaps and local resonance bandgaps, respectively. By modulating the local resonance bandgap, the Bragg bandgap can be tuned. The introduction of local bandgap increases the degree of freedom to modulate the transmission characteristics of MIM waveguides. Combining local resonant bandgap and Bragg bandgap can significantly increase the reflection bandwidth, achieve broadband filtering, and facilitate the miniaturization of waveguide devices.

Double-order coupling bandgap design of metamaterial beams and broadband vibration reduction properties

&lt;sec&gt;Local-resonance bandgap and Bragg bandgap can coexist in a metamaterial beam, and their coupling effect can be used to realize ultra-wide bandgap, which has great potential applications in the field of wide-band vibration reduction. Previous studies usually considered the single-order coupling between the local-resonance bandgap and Bragg bandgap in metamaterial beams with a single array of local resonators, which can only achieve the single-order ultra-wide coupling bandgap and cannot meet the wide-band vibration reduction requirements of double/multiple target frequency bands. In this paper, metamaterial beams with double arrays of local resonators are considered, and the regulation design and analysis of double-order coupling of local-resonance and Bragg bandgaps are carried out based on an analytical model of bending wave dispersion relation. Moreover, the vibration reduction characteristics of the double-frequency-resonator metamaterial beams with double-order coupling bandgaps are studied by using spectral element method and the finite element method. The main conclusions are as follows.&lt;/sec&gt;&lt;sec&gt;1) A design method is proposed for realizing double-order coupling wide bandgap in a metamaterial beam with double arrays of local resonators. By using this method, the resonance frequencies of the local resonators can be quickly designed on conditions that host beam parameters, lattice constant and added mass ratio of the local resonators are given.&lt;/sec&gt;&lt;sec&gt;2) The double-order coupling bandgaps in a metamaterial beam carrying double arrays of local resonators are compared with the single-order coupling bandgaps in metamaterial beams with a single array of local resonators. It is found that through proper design, the total normalized width of the double-order coupling bandgap can be much broader than that of the single-order coupling bandgap, so the double-order coupling bandgap is more beneficial to wide-band vibration reduction.&lt;/sec&gt;&lt;sec&gt;3) It is found that for a given total added mass ratio of the double arrays of local resonators, it is necessary to optimize the mass distribution ratio of the double resonators to maximize the total normalized width of double-order coupling bandgap. An approximate formula for designing the optimal mass distribution ratio of the double resonators is further established.&lt;/sec&gt;&lt;sec&gt;4) The spectral element method is used to study the vibration reduction characteristics of the metamaterial beams carrying double arrays of local resonators designed based on double-order bandgap coupling. The accuracy of the spectral element method is verified by comparing with the finite element method. The results show that significant vibration reduction can be achieved in two wide frequency bands corresponding to the double-order coupling bandgaps. The influences of number of unit cells and resonator damping on the vibration reduction characteristics of the metamaterial beam are further analyzed. It is shown that the increase of number of unit cells can enhance the vibration reduction performance in the bandgap, and the increase of resonator damping can effectively broaden the vibration reduction frequency band.&lt;/sec&gt;

Metamaterial-based absorbers for simultaneous absorption of air-borne sound and structural vibration

In this work, the idea of metamaterial-based absorbers is further extended to develop the concept of sound and vibration absorbers (SVA) capable of simultaneously absorbing sound waves propagating in air and structure. Each SVA unit cell consists of an outer coating and an inner core. The outer coating is a resilient closed-cell foam, superseding the commonly used open-cell porous materials to significantly improve the structural integrity and service endurance. The inner core embedded in the outer coating provides multidirectional inertia to introduce multi-modal interferences with the outer coating. Utilizing a combination of analytic and numerical techniques, we reveal the underlying physics of the SVA for the simultaneous absorption of sound and vibration. A proof-of-concept SVA sample was fabricated using an expanded polyethylene foam as the outer coating and a polylactide hollow cylinder as the inner core. Good agreements are observed between the simulated and experimental results in terms of the normal incident sound absorption coefficients and the frequency response functions. We find that the SVA exhibits two nearly-perfect sound absorption peaks at 850 Hz and 1210 Hz owing to the resonant-induced impedance matching. Meanwhile, the SVA can effectively attenuate structural vibration across an ultra-wide frequency range relying on the effects of the locally-resonant bandgap and the structural damping dissipation. Finally, the noise reduction feasibility of the SVA sample applied in a plate-cavity coupled system was demonstrated. Our reported findings open alternative perspectives for the construction of practical metamaterial-based absorbers to attenuate noise propagating in complex transmission paths.

Mitigating vibration and sound radiation with a digital piezoelectric meta-shell in heavy fluids

The vibration-induced sound radiation of shell structures has a significant influence on the acoustic-stealth performance of underwater vehicles. To tackle this challenge, a programmable meta-shell featuring periodically distributed piezoelectric shuntings is investigated in this research. Its tunable local-resonance (LR) bandgap, enabled by the digital shunting circuits, is utilized to mitigate the vibration and sound radiation of underwater shells. We establish an electrical-mechanical-acoustic coupling model for the meta-shell submerged in the unbounded heavy fluid. The quadratic velocity and the radiated sound power are computed to evaluate its global vibro-acoustic behaviors. It is demonstrated that, by strategically designing LR bandgaps around the wet-mode frequencies of meta-shells, both the vibration and the sound power could be effectively reduced at target modes. The impacts of electrical damping and poles on the vibro-acoustic reduction efficiency are also identified. Lastly, the experimental set-up of a piezoelectric meta-shell equipped with 48 independently controlled cells is built. The results achieved through underwater experiments validate the meta-shell's capability to programmably suppress vibro-acoustic response peaks associated with multiple wet modes.

Local resonance bandgap control in a particle-aligned magnetorheological metamaterial

Stimuli-responsive elastic metamaterials augment unique subwavelength features and wave manipulation capabilities with a degree of tunability, which enables them to cut across different time scales and frequency regimes. Here, we present an experimental framework for robust local resonance bandgap control enabled by enhanced magneto-mechanical coupling properties of a magnetorheological elastomer, serving as the resonating stiffness of a metamaterial cell. During the curing process, ferromagnetic particles in the elastomeric matrix are aligned under the effect of an external magnetic field. As a result, particle chains with preferred orientation form along the field direction. The resulting anisotropic behavior significantly boosts the sensitivity of the metamaterial’s elastic modulus to the imposed field during operation, which is then exploited to control the dispersive dynamics and experimentally shift the location and width of the resonance-based bandgap along the frequency axis. Finally, numerical simulations are used to project the performance of the magnetically-tunable metamaterial at stronger magnetic fields and increased levels of material anisotropy, as a blueprint for broader implementations of in situ tunable active metamaterials.

Broadening Low-Frequency Band Gap of Double-Panel Structure Using Locally Resonant Sonic Crystal Comprised of Slot-Type Helmholtz Resonators

An approach is presented to form and broaden the low-frequency band gap of the double panel structure (DPS) by using a locally resonant sonic crystal (LRSC) in this work. The LRSC is made of cylindrical Helmholtz resonators arranged on square lattice. Their designs are similar to a slot-type resonator, but have different depths of slot. Elongating the slit neck inward and distributing the depths of slots produce a broad local resonant band gap at low frequencies: an average insertion loss (IL) of 10.9 dB covering 520 Hz to 1160 Hz with a LRSC of 12 cm width. Next, the effect of porous material filled into the resonators on the local resonant band gap is evaluated. It is shown that filling of porous material into the resonators decreases the height and width of the local resonant band gap. Finally, the transmission losses (TLs) through the DPS with LRSC are calculated as a function of the incident angle of the sound wave for LRSC embedded in porous material and not. The results show that the porous material can be significantly reduce the incident angle dependency of TL through the DPS with LRSC.

Analysis of geometric defects in square locally resonant phononic crystals: A comparative study of modeling approachesa).

Phononic crystals can develop defects during manufacturing that alter the desired dynamic response and bandgap behavior. This frequency behavior change can enable successful defect inspection if the characteristic defect response is known. In this study, the behavior of a defective square unit cell comprising a freed and shortened leg is studied using a wave finite element method and an approximate continuous-lumped model to elucidate the defect induced qualitative dynamical features. These metrics are a computationally inexpensive alternative to modeling a defective unit cell within a large pristine array entirely in finite elements. The accuracy of these models is validated by comparing the result to a full finite element model. The impact of a shortened unit cell leg on the behaviors of an infinite array of defective cells and a finite array with a single defect are successfully predicted through dispersion curves and frequency response functions, respectively. These methods reveal defect-induced modes that split the local resonance bandgap of the pristine cell, as well as new anti-resonances resulting from the shortened leg. The study uses both approaches to evaluate the effect of defects in complex phononic crystal geometries and provides a comparative evaluation of the results of each model.

A new meta-rod attenuating low-frequency waves with external fixed supporting

To attenuate low-frequency waves in a rod, this study proposes a new meta-rod with compression springs attached to an external fixed supporting. In particular, an essential problem concerns how external constraints on the negative-stiffness system influence the bandgap properties and wave attenuation behaviour explored. We analytically derive the dispersion relation of the meta-rod and reveal its bandgap properties using the transfer matrix method. Then we develop a finite-length meta-rod model and investigate its wave attenuation behaviour by the finite element method. By attaching compression springs to an external fixed supporting, the bandgap is shifted to cover the lower frequencies range with deep wave attenuation. Furthermore, the local resonance bandgap and the Bragg bandgap are effectively coupled by tuning the stiffness of the compression springs to obtain a wider bandgap. We envision the proposed meta-rod can be used as an effective countermeasure to mitigate vibrations at low frequencies.

Vibration Characteristics of Metamaterial Periodic Multi-Span Beams with Elastic Supports

A novel metamaterial periodic multi-span beam with elastic supports and local resonators is designed and studied. The vibration characteristics of the designed metamaterial multi-span beam structure are investigated by the spectral element method (SEM) combined with the transfer matrix method (TMM). The accuracy and feasibility of the theoretical methods are validated by the finite element method (FEM) and vibration experiment. It is demonstrated that the metamaterial periodic multi-span beam with elastic supports can produce local resonance bandgaps in the low-frequency regions and Bragg bandgaps in the middle- and high-frequency regions. For some support stiffnesses, the amplitudes and widths of the bandgaps for the proposed structure are larger than those of the metamaterial multi-span beam with simply supported boundaries. Thus, the positions and ranges of the Bragg bandgaps can be adjusted by changing the stiffness of the elastic supports so as to improve the vibration reduction performance of the structure.

Rogue waves as modulational instability result in one-dimensional nonlinear triatomic acoustic metamaterials

In this study, a triatomic acoustic nonlinear metamaterial was modelled. It is a network of identical periodic unit cells made up of three different masses. These masses interact with connections modelled by springs whose stiffnesses are linear and nonlinear. Handling the motion equations of nth unit cell, the semidiscrete approximation method is used to extract the three-component coupled nonlinear Schrödinger equations. The dispersion relationship was determined. The results showed that the dispersion relation of the one-dimensional triatomic chain contained an acoustic branch and two optical branches. The dispersion curves show that the proposed acoustic metamaterial exhibits two local resonance bandgaps independent of spatial periodicity. The spectral range and frequency bandgap are related to the mass of the three atoms in the original cell. Based on the theory of linear stability analysis, modulation instability is the condition for the existence of rogue waves. Analytically, it is shown that modulational instability and rogue wave phenomena are strongly influenced by variations in the nonlinearity, dispersion, and diffraction terms, which contain the values of the masses and stiffnesses.

Quasi-static band gaps in metamaterial pipes with negative stiffness resonators

A novel configuration of periodically supported pipes with negative stiffness resonators (NSRs) at midspans is investigated in this paper. By leveraging the expanded bandwidth and enhanced damping effects offered by the NSRs, as well as the quasi-static (QS) band gaps of periodically supported pipes, this configuration enables the realization of ultrawide QS band gaps. The Floquet-Bloch theorem and Finite Element Method (FEM) are employed to determine the dispersion relationship of the infinite periodic pipe, elucidating the wave propagation characteristics. The band formation mechanism is revealed by conducting a comparative analysis with other systems in which unit cells only comprise periodic supports or NSRs. By analyzing the dispersion relationships, the effects of critical parameters on the transition and interplay between Bragg and local resonance (LR) band gaps are investigated. The results reveal that, with increasing absolute values of the negative stiffness ratio or mass ratio at a constant resonant frequency, the LR band gap created by NSRs eventually transforms into a wide QS band gap. The study further extends to metamaterial pipes of finite length, where FEM simulations are carried out using ANSYS to confirm the physical band-gap properties observed in the periodic systems. The advantage of the wave attenuation bandwidth introduced by NSRs is demonstrated by comparing the results with those obtained by traditional types of resonators. With the same added mass, the system with NSRs exhibits a substantially wider LR band gap than the traditional resonators. Finally, owing to the damping magnification mechanism of NS elements, when a reasonable amount of damping is introduced, the proposed system shows superiority in achieving ultrawide QS band gaps through band-gap merging over pipe systems with suspended resonators.

Tunable elastic wave modulation via local phase dispersion measurements of a piezoelectric metasurface with signal correlation enhancement

Tunable piezoelectric metasurfaces have been proposed as a means of adaptively steering incident elastic waves for various applications in vibration mitigation and control. Bonding piezoelectric material to thin structures introduces electromechanical coupling, enabling structural dynamics to be altered via tunable electric shunts connected across each unit cell. For example, by carefully calibrating the inductive shunts, it is possible to implement the discrete phase shifts necessary for gradient-based waveguiding behaviors. However, experimental validations of localized phase shifting are challenging due to the narrow bandgap of local resonators, resulting in poor transmission of incident waves and high sensitivity to transient noise. These factors, in combination with the difficulties in experimental circuitry synthesis, can lead to significant variability of data acquired within the bandgap operating region. This paper presents a systematic approach for extracting localized phase shifts by taking advantage of the inherent correlation between the incident and transmitted wavefronts. During this procedure, matched filtering greatly reduces noise in the transmitted signal when operating in or near bandgap frequencies. Experimental results demonstrate phase shifts as large as −170° within the locally resonant bandgap, with an average 28% reduction in error relative to a direct time domain measurement of phase, enabling effective comparison of the dispersive behavior with corresponding analytical and finite element models. In addition to demonstrating the tunable waveguide characteristics of a piezoelectric metasurface, this technique can easily be extended to validate localized phase shifting of other elastic waveguiding metasurfaces.

Programmable piezoelectric meta-rings with high-order digital circuits for suppressing structural and acoustic responses

In this paper, a programmable piezoelectric meta-ring equipped with high-order digital shunting circuits is designed for vibro-acoustic control. The structural and acoustic suppressing performance of the piezoelectric meta-ring in light fluid is tunable by re-programming transfer functions implemented in micro-controller units of the digital circuits. The radiated sound due to structural vibration of the meta-ring is attenuated in multiple local-resonance bandgaps, which is realized using high-order resonant circuits designed via the pole-zero placement method. An analytical model is developed for providing physical insights into vibro-acoustic behaviors of the piezoelectric meta-ring in light fluid. The analytical results agree well with numerical ones computed with the finite element method. The effects of electrical parameters such as gain parameter and electrical damping ratio on the sound radiation suppression of elastic rings are also discussed. To verify the programmability and high-order local-resonance bandgap performance of the piezoelectric meta-ring, the vibro-acoustic control experiment is carried out in the anechoic chamber. It is found that programmable bandgaps at different target frequencies of the piezoelectric meta-ring can be generated, which effectively suppress the structural–acoustic response peaks of the meta-ring.