Flexural Wave Band Gaps Research Articles

Isolation of flexural waves in a wide and low frequency range using a phononic beam with acoustic black hole configuration

We propose a phononic beam to realize flexural wave bandgaps in wide and low frequency regimes. A unit cell of the phononic beam consists of two uniform parts and a tapered part with acoustic black hole configuration sandwiched between uniform parts. We identify the effects of geometrical parameters of the ABH-based phononic beam on band structures and determine those geometrical parameters to obtain a bandgap from 3.6 Hz to 237.9 Hz with a lattice constant of 1/33 wavelength. For experimental realization, the displacement transmission of the phononic beam is measured by impact hammer test, and the result shows that low transmission of –35 dB to i–10 dB could be achieved in the wide bandgap.

Low-frequency broadband vibration attenuation of sandwich plate-type metastructures with periodic thin-wall tube cores

Sandwich structures are widely applied in modern industry such as aerospace, automobile as well as marine structures. However, the vibroacoustic properties of sandwich structures are adversely influenced by low effective mass. In this study, the flexural wave propagation characteristics and vibration mitigation performances of the periodic sandwich plate-type metastructures are investigated. The proposed sandwich plate-type metastructures are constituted of a sandwich plate with periodic thin-wall circular tube cores and periodically attached local stepped resonators. A finite element method combining Solid-Shell coupling numerical method and Bloch theory is presented to calculate the dispersion relations and the displacement fields of the eigenmodes of the infinite periodic sandwich plate-type metastructures. In addition, the acceleration frequency responses and vibration attenuation performances of finite periodic sandwich plate-type metastructures are numerically investigated and compared with the experimental measurements. Furthermore, the influences of geometric parameters on flexural wave band gaps are conducted. Results show that the sandwich plate-type metastructures can yield a low-frequency broad flexural wave band gap, in which the flexural wave propagation is conspicuously suppressed, resulting in significant flexural vibration attenuation. The flexural wave band gap and vibration attenuation performances can be effectively manipulated by designing geometric parameters of the sandwich plate-type metastructures.

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.

Complete sub-wavelength flexural wave band gaps in plates with periodic acoustic black holes

Acoustic Black Hole (ABH) effect shows promise for vibration control, but mainly limited to a relatively high frequency range. Though achievable in 1D periodic ABH structures, complete sub-wavelength band gaps (BGs) have not yet been realized in 2D configuration. Capitalizing on the unique wave propagation characteristics of the ABH, we propose a new type of plates containing periodically arranged double-layer ABH cells which offer complete and omnidirectional BGs. The phenomena originate from the combined effects of the ABH-specific local resonances and Bragg scattering, which are made possible through a dual process: a proper channeling of the wave propagation path and an impaired coupling between the ABH-induced local resonances and the global vibration of the unit cells. The former is warranted by a proper structural tailoring of the unit cells and the latter by the dynamics of the double-layer ABH design. It is shown that the BGs can be tuned through adjusting ABH parameters. Meanwhile, attaching the centers of the double ABH branches with a connecting cylinder can further broaden and lower the frequencies of the BGs as a result of the enhanced Bragg scattering. It is also demonstrated numerically and experimentally that remarkable vibration attenuation and energy insulation can be achieved in a plate with only a small number of ABH cells, thus pointing at the possibility of achieving sub-wavelength vibration control in structures with reasonable dimensions.

Coupled flexural-longitudinal waves in an origami metamaterial with uncoupled creases

Wave coupling exists in asymmetrical structures. In this work, we numerically and experimentally study wave coupling behaviors in one-dimensional origami metamaterials with uncoupled creases. Unlike conventional one-dimensional asymmetrical structures, the coupling characteristics in origami structures can be tuned and rich bandgap regulation can be achieved. By utilizing wave coupling, new flexural-wave bandgaps can be opened and shaped by folding. We propose a coupling ratio to explain different wave coupling phenomena with respect to the folding angles and structure parameters. Based on a proper realization of the creases and with two orthogonal fiber Bragging grating (FBG) displacement sensors, several experiments are performed to support our arguments on wave-coupling behaviors. Through this research, we also provide a systematic analyzing approach and an experimental method in studying coupling waves. Despite focusing on the crease-uncoupled origami structures, the provided multi-dimensional sensing technique enables one to study general origami metamaterials with coupled creases.

Flexural wave band gaps in a ternary periodic metamaterial plate using the plane wave expansion method

Low- and mid-frequency range vibrations usually present issues in their insulation. The development of periodic metamaterials (artificially engineered structures) has been thoroughly studied since their periodicity and geometric properties may yield good attenuation properties in such frequency ranges. Several works investigated periodic structures presenting binary or ternary material configurations using the Kirchhoff plate theory or binary material configurations using the Mindlin plate theory. In this paper, a ternary periodic metamaterials made of concentric circular inclusions of a rigid material coated with a soft material embedded in a matrix made of a low-cost recyclable polymer is investigated using the Mindlin plate theory combined with the plane wave expansion method. Computed band diagrams are compared to those obtained using the Kirchhoff plate theory and three-dimensional models. Additionally, the effect of different boundary conditions is investigated on finite structures using the finite element method. Our results assess the validity of the Mindlin plate model in such case and indicate its use in practical applications.

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.

Flexural wave propagation and vibration isolation characteristics of sandwich plate-type elastic metamaterials

Sandwich structures are widely used in the fields of aerospace, automobile as well as ship and offshore structures because of their excellent mechanical performances such as lightweight, high specific strength and high specific stiffness. In this study, the flexural wave band gaps and vibration isolation characteristics of sandwich plate-type elastic metamaterials are numerically and experimentally investigated. The proposed sandwich plate-type elastic metamaterials are constituted of local resonant stubs periodically deposited on a sandwich plate with periodic thin-wall aluminium tube cores. An efficient finite element method combined with a solid-shell coupling method and Bloch periodic boundary conditions is presented and validated by experimental measurements to calculate the dispersion relations and the acceleration frequency responses of sandwich plate-type elastic metamaterials. The influences of geometric parameters on the flexural wave band gaps are performed and discussed. Results show that the proposed sandwich plate-type elastic metamaterials can yield flexural wave band gaps in the low-frequency range and show significant performance on the flexural vibration isolation. Moreover, the flexural wave band gaps can be effectively modulated by the geometric parameters.

Improving sound insulation in low frequencies by multiple band-gaps in plate-type acoustic metamaterials

According to the mass law, low-frequency sound attenuation is very difficult in engineering because the mass and space of structures are usually limited. In this paper, we propose a plate-type acoustic metamaterial with good sound insulation in low-frequency ranges. By attaching multilayered rubber and metal cylinders on a thin plate, multiple flexural wave band-gaps at low-frequency ranges are obtained. The band structures and eigenmodes are elucidated by finite element calculations. Both the flexural wave transmission spectra and the sound transmission loss are investigated. The results show that the obtained multiple band-gaps lead to more than one sound insulation peak at low frequencies. The geometrical parameter effects on the band-gap frequencies are discussed. By combining cells with different geometrical parameters, the formed supercell can improve the low-frequency sound insulation efficiency and introduce much more variety. The results in this paper will be useful for sound and vibration control in engineering applications.

Investigation of flexural wave band gaps in a locally resonant metamaterial with plate-like resonators

This paper presents an investigation of flexural wave band gaps in locally resonant metamaterials (LRMs). An LRM is a periodic structure consisting of repeated unit cells containing a local resonator. Due to the local resonance occurring in the unit cell, the LRM induces a band gap (a frequency band in which no waves propagate). Discrete-like or beam-like resonators have generally been used to realise LRMs in previous research. By extending the beam-like resonator configuration, this paper studies LRMs with a plate-like resonator to exploit its advantages with respect to large design freedom. In order to understand flexural wave band gaps in an LRM with plate-like resonators, parametric studies are conducted with the development of a finite element model. Further, the influences of the plate-like resonator design parameters on flexural wave band gaps are investigated. Based on the parametric studies, the rules governing band gap properties are determined. Finally, tailoring flexural wave band gaps by adjusting the parameters is discussed.

Flexural wave band gaps in multi-resonator elastic metamaterial Timoshenko beams

Plane wave expansion (PWE) is one of the most commonly employed methods of calculating dispersion diagrams of phononic crystals. Nevertheless, the method has been used less frequently to deal with elastic metamaterials (EMs), such as beams and plates containing local resonators. Until now, PWE and extended plane wave expansion (EPWE) methods have not been used to calculate the dispersion diagrams of EM beams with multiple periodic arrays of attached multiple degrees of freedom (M-DOF) resonators. We propose PWE and EPWE formulations to analyse flexural wave band gaps in an EM Timoshenko beam with multiple periodic arrays of attached M-DOF resonators. These formulations can predict more accurate results at higher frequencies than previous formulations developed based on Euler–Bernoulli beam theory. A coupling between locally resonant and Bragg-type band gaps is observed considering two arrays of single degree of freedom (S-DOF) resonators. However, this coupling does not occur for a single array of 2-DOF resonators even if the mass ratio is not fixed. Multiple arrays of 2-DOF resonators achieve better attenuation performance than multiple arrays of S-DOF resonators at the first resonance frequencies. However, near the first Bragg frequency, the attenuation is higher for multiple arrays of S-DOF resonators. Mass ratio effect is also investigated and it is significant for design purposes. Multiple arrays of M-DOF resonators enlarge EM potential applications for vibration management.

Actively controllable flexural wave band gaps in beam-type acoustic metamaterials with shunted piezoelectric patches

In this paper, a tunable beam-type metamaterial composed of elastic base beams and periodically arrayed active beam-type resonators is designed and analyzed theoretically. The piezoelectric shunted array technique is applied to each engineered resonator, so that the band structure in the system can be actively controlled. Based on spectral element method, the dispersion relation of an infinite system as well as the transmission equation of a finite system are obtained explicitly. The effects of negative capacitance shunt and negative capacitance enhanced resonant shunt on band structures are considered numerically. It is shown that, on one hand, the negative capacitance shunts can sensitively control the widths and locations of local resonance band gaps. On the other hand, when a negative capacitance enhanced resonant shunt is applied, the enhanced meta-damping phenomenon emerges, leading to an extra wide and low-frequency band gap. The resistance in the shunt induces damping to the system, which together with the local resonance motion gives rise to meta-damping behavior. Such effect is further enhanced by negative capacitance, which increases the system electromechanical coupling property. Numerical simulations also show that such novel band gap can be realized only when the negative capacitance is close to the instability boundary and resistance in a certain region. In addition, the inductance of shunt can optimize the attenuation performance distribution in the extra broad band gap. Particularly, the transmission analyses show that the attenuation property of such extra wide band gap is as good as the general Bragg scattering band gaps. Such extremely wide and low-frequency band gap can be used for a wide range of engineering applications such as vibration suppression, sound absorption, acoustic filter, etc.

Using eddy-current vibration absorbers to design locally resonant periodic structures

Flexural wave bandgaps of traditional periodic structures are narrow and difficult to tune. Damping is an important parameter for controlling those bandgaps, but structural damping is also generally difficult to design and change, which is why there have been relatively few studies that explore the role of damping in bandgap design in the past. In the present paper, adjustable eddy-current vibration absorbers (ECVAs) are used as local resonators installed on a beam. The flexural wave bandgap and the vibration response of the beam with periodic ECVAs with different damping constants are analyzed, and how the damping influences the bandgap and the transmissibility is revealed. The experimental results agree well with the simulation results. The flexural wave is attenuated strongly within the bandgap, and the upper boundary of the bandgap rises gradually with the damping constant of the resonator, thereby widening the bandgap. However, the resonator damping influences the lower boundary of the bandgap only weakly. Moreover, if the periodic ECVAs are mistuned by a small amount and the resonator damping is small, then the structure has a number of separate locally resonant bandgaps. With a sufficient amount of eddy-current damping, the separate bandgaps become interconnected to form a larger bandgap. The present approach is a new way to broaden locally resonant bandgaps.

Flexural wave band gaps and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates

Vibrations in ship and offshore structures owing to various ocean environmental loads and excitations of power systems become increasingly serious. In this paper, the flexural wave propagation and vibration attenuation characteristics in periodic bi-directionally orthogonal stiffened plates are investigated. The dispersion relations and the displacement fields of the eigenmodes of infinite periodic bi-directionally orthogonal stiffened plates are calculated by using the finite element method in combination with Bloch periodic boundary conditions. Numerical results show that periodic bi-directionally orthogonal stiffened plates can yield complete and directional flexural wave band gaps, in which the propagation of flexural vibrational waves is prohibited and flexural vibration suppression is dramatically achieved. With the introduction of bi-directionally orthogonal stiffeners, the flexural wave and vibration energy is confined in the four corners of the plate owing to the scattering effect of the bi-directionally orthogonal stiffeners. The transmission spectra for a finite periodic stiffened plate are numerically and experimentally achieved to verify the existence of the flexural wave band gaps and vibration suppression characteristics. Furthermore, the effects of geometrical parameters on the flexural wave vibration band gaps are carried out. The flexural wave band gaps and vibration attenuation properties can be artificially modulated by changing the geometrical parameters of periodic bi-directionally orthogonal stiffened plates.

A tunable elastic metamaterial beam with flat-curved shape memory alloy resonators

When realizing an elastic metamaterial beam, beam-like resonators are regarded as the simplest forms of distributed resonators and their influences on the 1st flexural wave bandgap have been well studied. In this letter, to study the relation between the curvature of the beam-like resonators and the high-order bandgaps (specifically, the 2nd and 3rd bandgaps) and make the bandgaps tunable, we propose a metamaterial beam with a periodic array of two-way shape memory alloys (SMAs). The considered aging-treated Ni-rich Ti-Ni SMAs remember a curved shape with a central angle of 180° at high temperature (austenite phase, A phase) and a flat shape at low temperature (rhombohedral phase, R phase) without the bias springs required for conventional one-way SMAs. Our analyses show that, while reducing the performance of the 1st bandgap, the central frequency of the 2nd (3rd) bandgap of the metamaterial beam with flat-curved SMAs can be decreased by 30.1% (18.3%) with an increase in the bandgap width by 35.9% (19.7%) compared to that with the flat-flat SMAs. In addition, the first torsional mode-induced flexural bandgap is widened and lowered when using the 180° curved beam resonators. The proposed SMA-based metamaterial utilizing the two-way shape memory effect thus offers a flexible and diverse route for tuning the bandgaps.

Tunable flexural wave band gaps in a prestressed elastic beam with periodic smart resonators

This paper theoretically studies the propagation of flexural waves in an actively controllable locally resonant (LR) beam. It is shown that the band gaps can be simultaneously tuned by the axial force and the active electrical control actions. In some special cases, a super-wide pseudo-gap resulted from the combination of the resonance and Bragg gaps can be observed. The condition of inducing such pseudo-gap is further obtained with a closed-form expression with respect to the electrical control parameter and the axial force, which can be explored to actively control the broadband pseudo-gap by tuning these two parameters.

Flexural Wave Bandgap Property of a Periodic Pipe with Axial Load and Hydro-Pressure

A periodical composite material pipe is proposed based on the Bragg scattering mechanism of phononic crystals (PCs). The band gap (BG) properties of the flexural wave in this PC pipe under axial load and hydro-pressure are calculated using the transfer matrix (TM) method. The frequency response functions (FRFs) of the PC pipe under axial load and hydro-pressure are calculated using the finite element approach, and the mechanism is elucidated to illustrate the phenomenon. The results show that axial load and hydro-pressure and their combination all have a great influence on the flexural vibration properties. This research offers theoretical support for research on PC pipes with complex conditions and is of great significance in solving the problem of high pressure.

Flexural wave band gaps in metamaterial plates: A numerical and experimental study from infinite to finite models

The proposed numerical design approach investigates both infinite and finite models and includes an experimental validation of elastic wave absorption and vibration suppression for flexural wave band gaps in metamaterial plates. The resonant metamaterial is obtained using a 2-dimensional periodic array of resonators (mass-screws) mounted to a thin homogeneous plate. The sensitivity analysis of the band gap frequency range takes into account the uncertainties of all the design parameters of the metamaterial plate. The theoretical approach uses the finite element method (FEM) to compare the predicted band gaps, which are derived from infinite and finite models of the metamaterial. An automatic method is proposed to detect the frequency ranges of band gaps in the finite metamaterial based on the behavior of the corresponding bare plate. Directional plane wave excitation and point force excitation are applied to evaluate the efficiency of the detection method. The results of these analyses are compared with experimental measurements. The frequency ranges of experimental vibration attenuation are in good agreement with the theoretically predicted complete and directional band gaps.

Thermal stress effects on the flexural wave bandgap of a two-dimensional locally resonant acoustic metamaterial

The elastic wave bandgap is obviously affected by heat while considering thermal stress. Nevertheless, the flat band, occurring in the lowest flexural branch, has not yet been explained clearly. This study investigates the influence of thermal stress on a flexural wave bandgap in a two-dimensional three-component acoustic metamaterial. Simulation results demonstrate that the band structure shifts to a lower frequency range, and the vibration response appears at a larger amplitude due to the bending stiffness being softened by the compressive membrane force. In addition, the first flexural band reduces to zero frequency in the central Brillouin zone. By viewing the vibration modes of the proposed unit cell, it is found that the out-of-plane mode shape attenuates with increasing temperature, while the in-plane vibration modes are unaffected by thermal stress.

On natural frequencies of non-uniform beams modulated by finite periodic cells

It is well known that an infinite periodic beam can support flexural wave band gaps. However, in real applications, the number of the periodic cells is always limited. If a uniform beam is replaced by a non-uniform beam with finite periodicity, the vibration changes are vital by mysterious. This paper employs the transfer matrix method (TMM) to study the natural frequencies of the non-uniform beams with modulation by finite periodic cells. The effects of the amounts, cross section ratios, and arrangement forms of the periodic cells on the natural frequencies are explored. The relationship between the natural frequencies of the non-uniform beams with finite periodicity and the band gap boundaries of the corresponding infinite periodic beam is also investigated. Numerical results and conclusions obtained here are favorable for designing beams with good vibration control ability.