Flexural Band Gaps Research Articles

A low-frequency pure metal metamaterial absorber with continuously tunable stiffness

To address the incompatibility between high environmental adaptability and deep subwavelength characteristics in conventional local resonance metamaterials, and overcome the deficiencies in the stability of existing active control techniques for band gaps, this paper proposes a design method of pure metal vibration damping metamaterial with continuously tunable stiffness for wideband elastic wave absorption. We design a dual-helix narrow-slit pure metal metamaterial unit, which possesses the triple advantage of high spatial compactness, low stiffness characteristics, and high structural stability, enabling the opening of elastic flexural band gaps in the low-frequency range. Similar to the principle of a sliding rheostat, the introduction of continuously sliding plug-ins into the helical slits enables the continuous variation of the stiffness of the metamaterial unit, achieving a continuously tunable band gap effect. This successfully extends the effective band gap by more than ten times. The experimental results indicate that this metamaterial unit can be used as an additional vibration absorber to absorb the low-frequency vibration energy effectively. Furthermore, it advances the metamaterial absorbers from a purely passive narrowband design to a wideband tunable one. The pure metal double-helix metamaterials retain the subwavelength properties of metamaterials and are suitable for deployment in harsh environments. Simultaneously, by adjusting its stiffness, it substantially broadens the effective band gap range, presenting promising potential applications in various mechanical equipment operating under adverse conditions.

Numerical Investigation of Broad Mid-Frequency Flexural Bandgap in Composite Sandwich Structures with Periodic Hollow-Shaped Core Geometry

Numerical Investigation of Broad Mid-Frequency Flexural Bandgap in Composite Sandwich Structures with Periodic Hollow-Shaped Core Geometry

Low-frequency bandgap characteristics and vibration attenuation performance of metamaterial-tailored concrete-filled steel tube columns

Low-frequency vibrations, such as earthquakes and wind loads, may negatively affect the proper operation of buildings and bridges and even lead to structural failure. Even though, in the past two decades, metamaterial-based structures have been utilized in structure vibration suppression, they are mainly used in the mitigation of high-frequency vibrations, and it has been an open challenge to suppress low-frequency vibrations below 100Hz and ensure adequate bearing capacity in engineering structures. In this work, we present metamaterial composite column structures with built-in local resonators to achieve the low-frequency vibration mitigation effect. Compared to ordinary concrete-filled steel tube (CFST) columns, the metamaterial columns (meta-columns) can utilize the rubber shear modulus to create a low-frequency flexural bandgap in 30–50 Hz. Dynamic stiffness is introduced in the Timoshenko beam and Bloch's theorem, the metamaterial analytical model is established, and the bandgap generation mechanism and vibration reduction effect are analyzed in conjunction with numerical simulations. Then, the bandgap effect and vibration attenuation law of the meta-columns are investigated by shaker tests at different acceleration amplitudes. Guided by theory and simulations, we have successfully fabricated several meta-composite columns with low-frequency bandgaps, which attenuate up to half of the vibrational energy near the bandgap center. The bandgap ranges derived from the theoretical and numerical models show a high degree of consistency with the experimental results, with deviations generally within 10 %, all indicating that a bandgap exists near the resonator self-oscillation frequency. Additionally, the bandgap is significantly affected by the number of unit cells and resonators, especially the meta-column with fewer resonators fails to form a bandgap where the top response is smaller than the bottom response. As the number of resonators and metamaterial units increases, the meta-columns exhibit wider bandgap and stronger vibration reduction performance.

Broadband low-frequency sound insulation of a metamaterial plate with inertial amplification

The recent development of Inertia Amplification (IA) acoustic metamaterial (AMM) opens a new venue for insulating low-frequency sound waves. This differs from conventional approaches that rely excessively on the structural density of the insulator and, therefore, lead to prohibitively heavy structures. The IA-AMM acquires a large effective density and a broad flexural bandgap by amplifying the inertia of many small mass elements periodically distributed in its unit cells, thus holding great promise as an effective yet lightweight solution for low-frequency sound insulation. This paper explores this possibility through numerical and experimental investigations. A semi-analytical model of an IA-AMM plate is developed by combining analytical equations characterizing the kinematics of an IA mechanism and FEM equations governing the vibration of the host plate. Based on this model, sound transmission characteristics of the plate under various sound incidences are analyzed with different amplification angles. By comparing with a locally resonant AMM plate, we found that the IA-AMM plate's sound transmission losses (STLs) are significantly and systematically improved over a wide frequency range of 116–544 Hz. To understand the underlying physics, the band diagram of the IA-AMM plate is calculated, which indicates the formation of a flexural bandgap covering the frequencies of the improved STL. The velocity contour of the plate at the center frequency of this bandgap also reveals much-suppressed vibration velocity amplitudes and a dipole-like sound radiation directivity, further explaining the observed enhancement of the sound insulation ability. Numerically predicted improvement of sound insulation is finally confirmed by experiments.

Investigation on band gap mechanism and vibration attenuation characteristics of cantilever-beam-type power-exponent prismatic phononic crystal plates

This study proposes a novel phononic crystal (NPC) configuration composed of power-exponent prismatic phononic crystal (EPC) and cantilever beams to address the vibration problem of the ship plate structure. Through the dispersion curve, it is found that NPC has complete flexural band gaps of low-, medium-, and high-frequency. The medium-frequency and high-frequency band gaps are the coupled band gaps induced by local resonance under the superimposition between the resonance of cantilever beams and the energy concentration effect in the primary structure. It is found that the addition of cantilever beam can make EPC produce low-frequency, and medium-frequency band gaps and broaden high-frequency band gap. Changing the parameters of the cantilever beam can adjust the low-, medium-, and high-frequency band gaps at the same time. Changing the parameters of the power index prism can adjust the medium-frequency and high-frequency band gap without changing the low-frequency band gap. The present simulation and model tests validated the above conclusions. Compared with the traditional embedded ABH model, NPC cannot weaken the structural strength with greater flexibility in application, suggesting broad application prospects in ship structure engineering. The present research results can support the vibration control of ship structures.

Flexural Wave Propagation in a Double-Beam System Interconnected by Local Resonators with Two Degrees of Freedom

The flexural band gaps in a double-beam system on elastic foundations periodically interconnected by local resonators with two degrees of freedom were investigated using the plane wave expansion method. The transmission property of the finite periodic system was examined by the finite-element method to verify the existence of the band gaps. The mechanism of band gap formation was further studied according to the eigenmodes at the edges of band gaps and the transverse deformation patterns. The technique of broadening the band gaps was also proposed. The results indicated that significant attenuation still appears outside the band gaps for the heterolateral transmission due to the counteraction of the approximate symmetric and antisymmetric flexural bands. The band gaps can be effectively broadened, and the propagating waves are even completely eliminated in the last pass band by only adding a spring. It was also observed that increasing the distance between the left and right attached points and the gravity of the resonator gradually from the asymmetric distribution to the symmetric distribution makes two band gaps coupled to form a superwide gap. The presented results emphasize the promising potential of the double-beam system in controlling the propagation of the flexural wave.

Effect of Resonant Aggregate Parameters in Metaconcrete Thin Plates on Flexural Bandgaps: Numerical Simulations

Effect of Resonant Aggregate Parameters in Metaconcrete Thin Plates on Flexural Bandgaps: Numerical Simulations

Bandgap prediction for a beam containing membrane-arch-mass resonators

This work aims to propose a promising locally resonating system consisting of a tensioned elastic membrane and two-arch masses attached on the membrane surface. Traditional membrane-type resonators, which usually create one obvious attenuation zone at low frequencies, might not be efficient in multi-frequency vibration suppression. The proposed structure can produce an extra clear flexural attenuation region and shift bandgap frequencies below 300 Hz. By adjusting geometric parameters (thickness, width, and location) of the arch mass, the bandgap region can be tuned. Introducing a feasible analytical model for accurately predicting the first and second initial frequencies of the bandgaps for a beam structure containing membrane-arch-mass resonators is another focus of this study. The proposed theoretical framework can be used to tune the bandgap to different target frequency ranges without knowing the actual width of the bandgap. Finite-element analysis and experiments are conducted to verify the theoretical predictions. A good agreement is seen among the theoretical, finite-element analysis, and experimental results. In addition, adjacent cells with different arch-mass distributions can generate two pairs of flexural bandgaps, increasing the practicality in engineering applications. The proposed structure might be used in low-frequency vibration isolation and filters.

Analysis of flexural and torsional vibration band gaps in phononic crystal beam

Phononic crystals (PCs) play an important role in the reduction of vibration within a targeted frequency range. Most research of the one-dimensional PCs focuses on the properties or applications of the flexural band gaps; however, practically, the broadband environmental excitations also induce the torsional vibration. Hence, this paper studies a two-degrees-of-freedom local resonance (2DOF-LR) phononic crystal, which can both suppress the propagation of flexural and torsion waves in the same frequency range. To improve the computational efficiency, the Timoshenko theory with a modified transfer matrix (MTM) approach is used for flexural vibration band structure analysis. Through the finite element method (FEM), the vibration attenuation characteristics of the structure within the band gaps are verified. A comprehensive study is conducted to investigate the effects of geometric and material parameters on the attenuation characteristics of the phononic crystal. Finally, a prototype of the proposed phononic crystal is fabricated for the vibration experiment. It is worth noting that a novel experimental setup using the lever principle is designed in this paper to verify the torsional band gap. The frequency ranges of the experimental vibration attenuation match well with the numerical results.

Chiral trabeated metabeam for low-frequency multimode wave mitigation via dual-bandgap mechanism

An elastic wave in a physical beam naturally possesses many wave modes, such as flexural, longitudinal, and torsional. Therefore, suppression of all possible vibration modes has been rarely achieved in beam-shaped periodic systems, especially at low frequencies. Here we present a low-frequency complete bandgap mechanism by overlapping the flexural bandgap with the longitudinal-torsional bandgap. To strengthen the general framework, we enforce an extra degree of freedom (rotational and torsional-spring) on the spring-mass system for the flexural and coupled (longitudinal-torsional) modes. The low rotational stiffness provides a low flexural bandgap, whereas the torsional stiffness yields a coupled-mode bandgap. To meet these prerequisites in physical modeling, a chiral trabeated metabeam is conceived, which allows all wave modes to be suppressed by a complete bandgap. Apart from single-mode mitigation, our work provides a route to implementing a low-frequency complete bandgap in a periodic fashion, potentially enabling the use of chirality in elastic structures.

Wave Propagation in Tapered Periodic Curved Meta-Frame Using Floquet Theory

Abstract In this work, the elastic wave propagation and dispersion characteristics of a curved tapered frame structure are investigated analytically. Separately, wave propagation through uniform curved and straight tapered beam was reported in the existing literature; however, no literature reports the influence of simultaneous bent and taper on the wave propagation. In particular, the band characteristics for the curved and tapered beam with two types of cross sections, i.e., rectangular and circular, are presented. The paper elucidates that introducing a small periodic bent angle cross section produces a complete, viz., axial and flexural bandgap in the low-frequency region, and conicity enhances the width of the band. It is also evidenced that a curved tapered frame with a solid circular cross section induces a wider bandgap than the rectangular section. A complete first normalized bandwidth of 159% is achievable for the circular cross section and 123% in the case of the rectangular section. The complete result is presented in a non-dimensional framework for wider applicability. An analysis of a finite tapered curved frame structure also demonstrates the attenuating characteristics obtained from the band structure of the infinite structure. The partial wave mode conversion, i.e., generation of coupled axial and flexural mode from a purely axial or flexural mode in an uncoupled medium, is observed. This wave conversion is perceived in reflected and transmitted waves while this curved tapered frame is inserted between the two uniform cross-sectional straight frames.

Flexural band gaps and vibration control of a periodic railway track

Periodic structures exhibit unique band gap characteristics by virtue of which they behave as vibro-acoustic filters thereby allowing only waves within a certain frequency range to pass through. In this paper, lateral and vertical flexural wave propagation and vibration control of a railway track periodically supported on rigid sleepers using fastenings are studied in depth. The dispersion relations in both lateral and vertical directions are obtained using the Floquet-Bloch theorem and the resulting dispersion curves are verified using finite element models. Afterwards, tuned mass dampers (TMDs) with different mass ratios are designed to control vibrations of the examined rail in both the directions. Moreover, the influence of damping of rail and resonators on band gap characteristics is investigated. As a replacement to the conventional TMD, a novel possibility to control vibration relies on using another existing rail as a lateral distributed resonator (LDR). Although the effectiveness of LDR is lower than that of localized resonators, the former represents a simple and promising way to control vibrations. Efficacy of the proposed control methods is finally verified by applying a random Gaussian white noise input. The study presented here is useful to understand the propagation and attenuation behavior of flexural waves and to develop efficient and novel vibration control strategies for track structures.

Modal analysis of flexural band gaps in a membrane acoustic metamaterial (MAM) and waveguides affected by shape characteristics

This article explains the propagation of flexural waves of membrane acoustic metamaterials and waveguide effects from the perspective of modal analysis. The flexural band gaps are calculated with both single and double periodic Bloch-Floquet boundary conditions. The unit cells with one and two masses attached to the membrane are applied to get the transmittance of the flexural wave on a beam/plate. Local resonance and Bragg scattering mechanism is analyzed in detail through mode cloud image. The topological waveguide plates are designed based on structural asymmetry. The proposed MAM waveguide is strongly related to the rotation angle of the attached dual-mass. Finally, the dispersion relation of the supercell indicates the energy dissipation mechanism in the waveguide path.

Low frequency ship vibration isolation using the band gap concept of sandwich plate-type elastic metastructures

Elastic metamaterial is a type of artificial composite material/structure with unique subwavelength physics property, which shows attractive potential application in low-frequency sound and vibration control. In this paper, a ship vibration isolation method using the band gap concept of sandwich plate-type elastic metastructures is proposed to solve the low frequency vibration and noise control of marine power system. The flexural wave propagation and low-frequency vibration isolation characteristics of sandwich plate-type elastic metamaterials are investigated by using the finite element method combination with Solid-Shell multi-physics coupling model and Bloch theory. Numerical results demonstrate that the proposed sandwich plate-type elastic metastructures possess significant low-frequency subwavelength flexural wave band-gap and vibration isolation characteristics under the premise of structural strength and stiffness. The formation mechanism of flexural band gap is mainly attributed to the mode coupling of the local resonant mode and the Lamb wave mode of sandwich plate. The experimental measurements of flexural vibration transmission spectra were conducted to validate the accuracy and reliability of the numerical calculation method. In addition, the influences of geometrical parameters on the flexural wave band gap are studied. On this basis, the potential application of the proposed ship vibration isolation method in the marine power system is explored.

An Inertant Elastic Metamaterial Plate With Extra Wide Low-Frequency Flexural Band Gaps

Abstract Arranging inerter arrays in designing metamaterials can achieve low-frequency vibration suppression even with a small configuration mass. In this work, we investigate flexural wave bandgap properties of an elastic metamaterial plate with periodic arrays of inerter-based dynamic vibration absorbers (IDVAs). By extending the plane wave expansion (PWE) method, the inertant elastic metamaterial plate is explicitly formulated in which the interactions of the attached IDVAs and the host plate are considered. Due to the additional degree-of-freedom induced by each IDVA, multiple band gaps are obtained. Along the ΓX direction, the inertant elastic metamaterial plate exhibits two locally resonant (LR) band gaps and one Bragg (BG) band gap. In contrast, along the ΓM direction, two adjacent LR band gaps are obtained. Detailed parametric analyses are conducted to investigate the relationships between the flexural wave bandgap properties and the structural inertant parameters. With a dissipative mechanism added to the IDVAs, extremely wide band gaps in different directions can be further generated. Finally, by adopting an effective added mass technique in the finite element method, displacement transmission and vibration modes of a finite inertant elastic metamaterial plate are obtained. Our investigation indicates that the proposed inertant elastic metamaterial plate has extra-wide low-frequency flexural band gaps and therefore has potential applications in engineering vibration prohibition.

Design of a local resonator using topology optimization to tailor bandgaps in plate structures

The main aim of this paper is to present a topology optimization method for a local resonator, in order to tailor flexural bandgaps in plate structures. The local resonator is required to be designed systematically in order to tailor the bandgap at the desired frequency range. In previous studies, beam or plate-like resonators were usually used for the design purpose, but the freedom of design is limited particularly when the allowable dimensions are highly limited. Regarding the design issue, this research presents a systematic method using topology optimization to design the local resonator with the maximized bandgap at the target frequency. A plate-like resonator is parameterized to fully exploit the allowable design space of a unit cell. The topology optimization is carried out with the non-gradient based algorithm i.e. simulated annealing (SA). To assist the SA, which requires a large number of function evaluations, two computationally efficient finite element (FE) approaches, namely plate modelling and the reduced Bloch mode expansion, are combined with the topology optimization. In addition, numerical techniques are applied to solve issues arising in the design process (i.e. design disconnection, hinged design, and design redundancy). Numerical examples demonstrate the effectiveness of the presented method for creating bandgaps at frequencies below 500 Hz. The results exhibit increases in vibration isolation up to 20 dB at the designed frequencies.

Flexural band gaps and response attenuation of periodic piping systems enhanced with localized and distributed resonators

Novel metamaterial concepts can be used to economically reduce flexural vibrations in coupled pipe-rack systems. Here, we model pipe on flexible supports as periodic systems and formulate dispersion relations using Floquet-Bloch theory which is verified by a finite element model. Owing to the flexibility of the coupled system, a narrow pass band is created in low frequency regime, in contrast to the case of pipe without any rack. Two types of vibration reduction mechanisms are investigated for pipe with different supports, i.e. simple and elastic support. In order to tune the band gap behaviour, lateral localized resonators are attached at the centre of each unit cell; conversely, the lateral distributed resonators are realized with a secondary pipe existing in the system. The results reveal that both Bragg and resonance type band gaps coexist in piping systems due to the presence of spatial periodicity and local resonance. Although, the response attenuation of a coupled pipe-rack system with distributed resonators is found to be little lower than the case with the localized one, the relatively low stiffness and damping values lead to cheaper solutions. Therefore, the proposed concept of distributed resonators represents a promising application in piping, power and process industries.

Elastic metamaterial shaft with a stack-like resonator for low-frequency vibration isolation

This paper presents an elastic metamaterial shaft with a new class of stack-like resonators consisting of bonded periodically annular rubber and lead rings that support the formation of ultra-low-frequency vibration band gaps because of reduced stiffness resulting from the discretized rubber rings. By using finite element method, the band structures and the transmission power spectra of the proposed metamaterial shaft are investigated. Subsequently, the mechanism of the band gaps is illustrated by analyzing the displacement fields of extracted mode shapes at the edge of band gaps. Lastly, finite element method is used to explore the effects of geometrical parameters on the band gaps. The related results confirm the very-low-frequency vibration mitigation performance of the proposed meta-shaft below 120 Hz and indicate there is a complete bandgap in which flexural, longitudinal and torsional vibration can be simultaneously attenuated. Moreover, multiple flexural band gaps are observed in the concerned region of frequencies in our design, and the mechanism behind this characteristic is analyzed by eigenmodes. The new structure presented here can thus be used to reduce vibrations of shaft-like structures at low frequencies in the practical environment.

A Numerical Method for Flexural Vibration Band Gaps in A Phononic Crystal Beam with Locally Resonant Oscillators

The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.

Mid-frequency band gap performance of sandwich composites with unconventional core geometries

In this work novel unconventional core architectures are presented which are able to induce flexural band gaps while not being detrimental for structural bending stiffness of the sandwich structures. Two different core schemes are examined with both of them exhibiting low-frequency stop bands. While unconventional, the designs of the core offer a novel solution which can be easily manufactured in high volume parts using two-dimensional automated cutting machine. A hybrid finite element and periodic structure theory scheme is employed for the calculation of the stiffness and mass matrices, and periodic structure theory is used to obtain the wave propagation of the beams. Having acquired the wave dispersion curves and the finite element analysis’ results, two specimens are manufactured using carbon fibre cured plates and commercially available PVC foam as core material. Experimental measurements of the dynamic performance of the structures are conducted using a laser vibrometer and electrodynamic shaker setup.