Flexural Wave Band Gaps Research Articles

Ultra-low frequency and broadband flexural wave attenuation using an inertant nonlinear metamaterial beam

Attenuation of elastic waves in the extremely low frequency range is a considerable challenge. While linear acoustic metamaterials can manipulate elastic waves, their effectiveness is often limited to a narrow frequency range. The present paper proposes a novel inertant nonlinear metamaterial beam to address the challenging problem of the ultra-low frequency broadband flexural wave attenuation. The amplitude-frequency response and dispersion relations of the flexural waves are obtained using the first-order harmonic balance method, where four different resonant units are investigated and the resulting band gaps are compared. Finite element (FE) simulations are also conducted to evaluate the transmission spectra of the flexural vibration through a finite metamaterial beam. The good consistency between the band gaps predicted by the analytical model and the FE simulation verifies the correctness and effectiveness of the developed analytical approach. The results of this study demonstrate that introducing the inertance and softening nonlinearity into the resonant units can significantly widen the flexural wave band gaps within the low-frequency range. This finding provides a viable solution for engineering applications that require low-frequency vibration suppression and isolation.

Broadband frequency vibration isolation in pipelines using BCC meta-lattice sandwich sleeves

Pipeline structures usually suffer from serious vibration problems due to pressure pulsation and fluid excitation, potentially compromising their structural stability and service life. In this paper, a lightweight body center cubic (BCC) meta-lattice sandwich sleeve is proposed to isolate vibrations in pipelines within a broadband frequency range. The vibration isolation is based on the concepts of the locally resonant metamaterials. An equivalent mass-spring model is conducted to theoretically predict frequency range and elucidate the generation mechanism of the band gap. In addition, an analytical methodology is also proposed to predict the equivalent stiffness of the BCC meta-lattice sleeves. The influence of geometric parameters and configuration layout of the meta-lattice on the flexural wave band gap is further investigated. Numerical studies and experimental tests are also conducted to validate the vibration isolation effectiveness, considering various pipe lengths and different numbers of meta-lattice units. The proposed design provides fundamental support for the design of lightweight vibration isolation structure for pipelines.

Multifunctional metamaterials for simultaneous sound absorption and flexural vibration isolation

Controlling the noise and vibration simultaneously in some engineering scenes is extremely essential to eliminate harmful effects on the humans and equipments. To overcome this challenge, a multifunctional metamaterial plate with both airborne sound absorbtion and flexural vibration isolation simultaneously is investigated theoretically, numerically, and experimentally. The plate consists of a host plate and a Helmholtz resonator, where Helmholtz resonance is used to absorb sound waves and its external solid domain plays as role of a local resonator to isolate the flexural vibrations. The theoretical models were established to characterize the mechanism of the sound absorption and vibration isolation. The experimental results demonstrated that the proposed metamaterial plate can achieve perfect sound absorption at 649 Hz and a flexural wave band gap at 289–470 Hz. Besides, although the sound absorption and vibration isolation functions are integrated in the same component, our theory identifies several critical geometrical parameters which can manipulate the absorption peak and the flexural wave bandgap, allowing us to design its frequency of multiple functions as desired. This work provides a promising solution for noise and vibration reduction in engineering.

Low-Frequency Flexural Wave Acoustic Insulation Characteristics of a Double-Layer Metamaterial Plate

The effects of geometric parameters and constituent material parameters on the flexural wave bandgap of a locally resonant double-layer metamaterial plate are investigated using the finite element method. Reasonable parameters are obtained through optimization; then, a double-layer plate made of ABS material with added aluminum mass blocks is designed. The results show that it can open the low-frequency (112--124 Hz) flexural wave bandgap at a subwavelength size. The metamaterial plate is modeled as an equivalent spring-mass system, and the frequency range with negative density for the equivalent dynamic mass is consistent with the width of the flexural wave bandgap. The bandgap generation mechanism is further elucidated. Sound insulation performance of double-layer metamaterial panels verified in an acoustic chamber. The maximum amplitude of STL is achieved at the 112 Hz point in the bandgap frequency range, which indicates that the low-frequency bandgap can effectively reduce vibration and noise. These results demonstrate that this novel double-layer metamaterial plate has wide applications in engineering.

Low-frequency vibration bandgaps and deep learning-based intelligent design method of Y-shaped core sandwich metabeams

In this paper, the wave propagation mechanics theoretical model of Y-shaped core sandwich metabeams with local resonators is established based on the spectral element method, and the flexural wave bandgaps and vibration isolation characteristics are studied. The reliability of the theoretical model of Y-shaped core sandwich metabeams is verified by the finite element method and experiment. On this basis, a deep learning-based intelligent design method for predicting the vibration transmission characteristics and structure design of Y-shaped core sandwich metabeams is proposed. A dataset is created using the theoretical model of wave dynamics, and deep neural networks are constructed to forward prediction of transmission characteristics and intelligent design of Y-shaped core sandwich metabeams, respectively. Results indicate that the bandgap range of Y-shaped core sandwich metabeams is 2.5 times that of Y-shaped core sandwich beams, and the start frequency of bandgap is reduced by 578 Hz. The prediction of the transmission characteristic curve is in good agreement with the target, and the average relative error of intelligent design is below 3 %, which verifies the precision of the intelligent design method. The innovative, intelligent design method provides a novel way to realize engineering vibration reduction design of acoustic metamaterials rapidly.

Flexural vibration suppression behavior of sleeved phononic crystal pipes in thermal environment

Design idea of phononic crystal has been introduced into vibration suppression of pipes and presents expected effects. During service period, thermal environment is a common factor that may change the dynamic behavior of system. In this work, thermal influences on wave attenuation effect are studied for the single-component PC pipe which is constructed by attaching sleeves to the original bare pipe. Under the preloading effect of thermal stresses, flexural wave bandgap shifts toward the lower frequency range with temperature increment. Bandwidth gets wider for pipes with sleeves of a relative length larger than 0.5, while the bandgap narrows down or even closes up for pipes with shorter sleeves. The opposite variation trends are caused by the reversal of edge modes and the unequal responses of the two edge frequencies to thermal stress stiffness. Deeper reason is the discrepancy in determining factor for frequency between the two edge modes. As a result, the two edge frequency loci drop asynchronously under thermal load, and the transition point of bandgap shifts leftward. A thermo-vibrational joint test system is constructed and flexural wave transmittance is measured, for the first time, for a heated PC pipe. Experimental phenomena and simulation results suggest that the initial deflection and non-ideal constraint of the specimen make the attenuation band present opposite variation trends compared to the ideal model. The former strengthens the stiffness of the PC pipe, and the latter weakens the thermal influence on the PC pipe. Both the two non-ideal factors have a more significant impact on starting frequency of the attenuation band due to the difference in sensitivity of bandgap edge frequencies to stiffness perturbation of the PC pipe.

Flexural wave bandgaps of corrugated-core sandwich metabeam with inertial amplified resonators using the spectral element method

In this paper, we present a theoretical investigation of flexural wave propagation and vibration bandgap properties in corrugated-core sandwich metabeam with inertial amplified resonators. The complete analytical solutions for the flexural wave propagation with free boundary conditions are obtained through the spectral element method. Finite element calculation of transmission characteristics of sandwich metabeam with inertial amplified resonators are carried out to validate the accuracy of the proposed complete analytical solutions. In addition, the vibration reduction effect of sandwich metabeam with inertial amplified resonators is compared with that of the corrugated-core metabeam with local resonance. The influences of system parameters on the flexural wave attenuation are analyzed. Results show that the complete analytical solution for the proposed sandwich metabeam structure can be effectively used to predict the vibrational properties. In addition, significant enlargement of low-frequency bandgap can be obtained by using an inertial amplification sandwich metabeam structure. The proposed sandwich metabeam structure has good bandgap adjustment ability.

Controllable flexural wave bandgap in extensible metamaterial beams with embedded multiple resonators

Controllable flexural wave bandgap in extensible metamaterial beams with embedded multiple resonators

Phononic crystal pipe with periodically attached sleeves for vibration suppression

Flexural vibration of pipes is a common problem in many practical engineering applications, which usually influences the operating state of the system or even causes damage of the equipment. In the present work, an easy-realized design of phononic crystal pipe is proposed to implement vibration suppression by directly upgrading the bare pipe with attached sleeves without replacing the original structure. Based on the mechanism of Bragg scattering, flexural wave bandgap could be obtained with the periodical changing in cross section of the sleeved pipe. Sleeve installation generates both added mass and added stiffness simultaneously, which leads to the parabolic variations in edge frequencies of the bandgap against sleeve dimensions. Asynchronous recovery of the two frequency loci makes the bandgap close up when the sleeve covers about half length of the unit cell. This feature is not observed in analyses of binary phononic crystal pipes. Equivalent models are proposed to discuss the predicted variations, and the singular dependence of closure of bandgap on sleeve length is theoretical explained. Experiments are carried out to validate the feasibility of attaching sleeves for flexural vibration suppression of pipes. The proposed design could effectively attenuate the vibration energy with the intended bandgap.

Flexural Wave Bandgaps in a Prestressed Multisupported Timoshenko Beam with Periodic Inerter-Based Dynamic Vibration Absorbers

Locally resonant (LR) metamaterial structures possess bandgaps in which wave propagation is significantly attenuated. In this paper, we discuss flexural wave bandgaps in an LR beam subjected to a global axial force and multiple vertical elastic supports. An array of inerter-based dynamic vibration absorbers (IDVAs) was periodically attached to the LR beam. The flexural wave band structure of this prestressed multisupported LR beam was first derived using the transfer matrix method (TMM) and then explicitly illustrated through a numerical example. Four bandgaps were identified: a bandgap located in the low-frequency zone, a Bragg band generated by Bragg scattering, and two LR bands generated by the local resonance of the IDVAs. The effects of the IDVA parameters, axial force, and vertical elastic support on the properties of the bandgaps were evaluated. In particular, the bandgaps merged accompanied by an exchange of their edge frequencies. The bandwidth of the merged bandgap was nearly equal to the sum of the bandwidths of the bandgaps involved, indicating a method for controlling broadband flexural vibration through the bandgap splicing mechanism.

Inverse Design of Micro Phononic Beams Incorporating Size Effects via Tandem Neural Network

Phononic crystals of the smaller scale show a promising future in the field of vibration and sound reduction owing to their capability of accurate manipulation of elastic waves arising from size-dependent band gaps. However, manipulating band gaps is still a major challenge for existing design approaches. In order to obtain the microcomposites with desired band gaps, a data drive approach is proposed in this study. A tandem neural network is trained to establish the mapping relation between the flexural wave band gaps and the microphononic beams. The dynamic characteristics of wave motion are described using the modified coupled stress theory, and the transfer matrix method is employed to obtain the band gaps within the size effects. The results show that the proposed network enables feasible generated micro phononic beams and works better than the neural network that outputs design parameters without the help of the forward path. Moreover, even size effects are diminished with increasing unit cell length, the trained model can still generate phononic beams with anticipated band gaps. The present work can definitely pave the way to pursue new breakthroughs in micro phononic crystals and metamaterials research.

A nonlinear acoustic metamaterial beam with tunable flexural wave band gaps

Although linear acoustic metamaterials (LAMs) have been widely studied, the design and analyses of nonlinear acoustic metamaterials (NAMs) are relatively immature. In this paper, a new NAM beam is designed, which is composed of a homogeneous beam with periodic resonant units attached on its top surface. Each resonant unit consists of two Duffing oscillators connected in series. First, an analytical model is developed based on the harmonic balance method, from which the amplitude-frequency response and dispersion of flexural wave are derived, and the position and width of the band gaps produced by this novel structure are quantitatively evaluated. Then, the effects of the linear and nonlinear stiffness, as well as the arrangement of the series-connected resonators on the frequency band structure are discussed. A finite element (FE) simulation is conducted to obtain the transmission of a finite NAM beam. The numerical results show that the frequency range of vibration suppression is consistent with the analytically predicted band gap, which validate the developed analytical model. Our study demonstrates that comparing with the corresponding linear counterpart, the system nonlinearity effectively enhances the tunability of the band gap of the flexural wave. It’s worth mentioning that an ultra-wide band gap starting from 0Hz can be achieved via introducing softening nonlinearity, which is quite attractive in engineering applications requiring low-frequency vibration suppression.

Broadband low-frequency flexural wave attenuation in beam-type metastructures with double-sides inertial amplified resonators

We present the theoretical investigation on the flexural wave propagation and vibration attenuation characteristics in a Euler–Bernoulli beam-type metastructure with double-sides inertial amplified resonators. Based on Bloch theory, the dispersion relation and propagation characteristics of flexural waves are calculated by the spectral element method. The influences of system parameters on the flexural wave attenuation of the proposed beam-type metastructures are analyzed. Results show that broadband low-frequency flexural wave attenuation can be achieved by the proposed double-sides beam-type metastructures. The tunability of flexural wave band gap and vibration suppression performance significantly enhances. In addition, the low-frequency flexural wave band gap can be effectively modulated and optimized by changing the system parameters.

Flexural Wave Band Gaps in Periodic Bi-Directionally Orthogonal Stiffened Plates with Holes

Plate frame structure with holes is a novel candidate to reduce the weight and strengthen the architectural structures to fulfill the requirements of practical engineering applications. The designed periodic hole structure can generate a band gap from the perspective of the frequency spectrum. In this study, a new band gap calculation method was proposed under the basic energy method framework for the analysis of hole periodic structures. Gaussian wavelet function with a good localization characteristic was adopted to overcome the inaccurately assumed displacement filed function due to the introduced hole issue. The entire plate structure was divided into cell elements and an artificial spring model was introduced to satisfy the cell coupling conditions. The coupling constraints were converted into elastic potential energy to be added to the energy functional for the entire system. The dispersion curves obtained by this method are in excellent agreement with the results of the finite element method. It was observed that periodic hole plates lead to the generation of complete and directional flexural wave band gaps. By introducing bi-directionally orthogonal stiffeners, the band gap of periodic hole plates could be significantly lengthened and complete band gaps could be successfully generated. Furthermore, geometrical parametric studies on flexural wave band gaps were analyzed. A wide and complete band gap could be realized by increasing the width of the stiffener to a certain extent, but not fully proportional.

Study on broad flexural wave bandgaps of piezoelectric phononic crystal plates for the vibration and noise attenuation

This paper explores a new approach to reducing the interior noise of one vehicle body through the flexural wave bandgap. The phononic crystal is designed by attaching the periodic piezoelectric stacks with the shunting circuits on the host plate. To better implement the frequency-dependent properties, the theoretical model of the piezoelectric stack is improved through capacitance identification. On this basis, the plane wave expansion method is employed to calculate the bandgap. An ultra-wide bandgap is achieved by tuning the shunting circuits and confirmed via the finite element method. The influence of electrical parameters on flexural wave attenuation is investigated. The applicability of the presented resonators for different host plates under the presented design guideline is examined. Subsequently, the concept of phononic crystal with the unique bandgap is exploited to enhance the interior sound performance of one vehicle body. Combining the sound response analysis with the panel acoustic contribution analysis, the numerical study is performed to pick out the critical panels for the arrangement of resonators. Then a circuit control strategy is proposed to activate the wide bandgap. The results demonstrate that the present piezoelectric phononic crystal plate has a significant advantage in hindering the flexural wave propagation and suppressing structure-borne noise over a broadband low-frequency range.

Band Gaps and Vibration Attenuation Characteristics Analysis in Homogeneous Beam Coupled With Periodic Oscillators Based on the Method of Reverberation-Ray Matrix

A periodic beam-oscillators coupling system is proposed as a physical model in this paper for analyzing the dynamic characteristics of periodic support beams and low-frequency flexural wave vibration of slender stiffened plate structures. The dispersion relation of flexural wave in the infinite long homogeneous beam coupled with periodic oscillators is calculated using the method of reverberation-ray matrix combined with the Bloch theorem. The accuracy and effectiveness of the method of reverberation-ray matrix in analyzing the band gaps and vibration characteristics of the homogeneous beam coupled with periodic oscillators are verified by the numerical results of the finite long homogeneous beam coupled with periodic oscillators. Both the analytical and numerical results show the existences of flexural wave band gaps in the homogeneous beam coupled with periodic oscillators, in which the propagation of the flexural waves is prohibited and flexural wave vibration is significantly suppressed. Furthermore, the effects of structural and material parameters on the flexural wave band gaps characteristics are respectively investigated. The flexural wave band gaps can be adjusted and optimized manually by adjusting structural and material parameters, which can be applied to vibration and noise control design of periodic support beams and slender stiffened plate structures.

Flexural wave band gaps of steel bridge decks periodically stiffened with U-ribs: Mechanism and influencing factors

To effectively and efficiently control vibrations in steel bridge decks, the flexural wave band gap and vibration attenuation mechanism are studied. The band structures and displacement fields of the eigenmodes at the band gap edges of infinite periodic U-rib stiffened plates are calculated. The calculation method is combining the finite element method with Bloch periodic boundary conditions. A scaled specimen test of a finite periodic U-rib stiffened plate is used to validate the numerical results. A comparison of results confirms that introducing the U-rib allows the periodic U-rib stiffened plates to yield several flexural wave band gaps. In the band gaps, flexural wave propagation is stopped and clear flexural vibration suppression is achieved. Furthermore, the effects of geometric parameters and structural schemes on the flexural wave band gaps are analyzed in detail. Finally, by analyzing a specific case, it is demonstrated that the flexural wave band gaps and vibration attenuation properties can be controlled. This provides a new approach to vibration and noise control in steel bridges.

Flexural wave bandgap properties in metamaterial dual-beam structure

In this paper, a metamaterial dual-beam structure is firstly constructed by periodically adding pillars between two identical homogeneous beams. By simplifying the pillar as an equivalent spring-mass-spring system, another metamaterial dual-beam structure carrying periodic spring-mass-spring resonators is then established. This equivalent representation method is validated through both a vibration transmission analysis and a band structure analysis. Two bands corresponding to the antisymmetric and symmetric flexural vibration modes respectively open a flexural narrow bandgap in the low frequency range. Then, the improved structure with high-static-low-dynamic stiffness (HSLDS) resonators is proposed in the need of widening the bandgap and lowering the beginning frequency. Results show that with the decrease of the proportion of the residual stiffness, the beginning frequency of the bandgap can be reduced, while the ending frequency keeps unchanged. This results in a relative bandgap as high as 1.465, which is about 4.4 times of the original structure.

Propagation Characteristics of Flexural Wave in One-Dimensional Phononic Crystals Based on Lattice Dynamics Model

In this paper, we establish discrete flexural lattice chain models of Bragg and locally resonant phononic crystals by setting mass defect atoms and local resonant elements on the flexural lattice chain. The bandgap characteristics of flexural wave in phononic crystals are studied by establishing the governing equations of the model. The results from models show that with the change of the mass ratio of defective atoms to normal atoms, the bandgap of the flexural wave produced by Bragg scattering shows a certain rule. When the local resonant bandgap and Bragg scattering bandgap are close to each other, the two bandgaps will be coupled to form a wider flexural wave bandgap. The effect of axial strain on bending wave propagation is only the shift of bandgap position. The effect of material damping on the propagation of a bending wave is only energy dissipation at high frequency. In addition, we use finite element simulation to calculate the bandgap of flexural wave in phononic crystals with mass defects, and the results are consistent with lattice chain model. This shows that lattice chain model can effectively guide the bandgap design of phononic crystals. This comprehensive study may help to elucidate the rule of bandgap generation of flexural wave in one-dimensional phononic crystals.

Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator

This paper aims to present a theoretical model of flexural wave bandgaps in locally resonant cylindrical curved shell and investigates a technique for broadband wave mitigation using a metamaterial cylindrical curved shell. Firstly, cartesian coordinates are converted to cylindrical coordinates in the cylindrical curved shell structure to construct periodic conditions in the circumferential direction. Then, governing equations are derived by using the extended Hamilton principle, and plane wave expansion (PWE) method is implemented to calculate the band structures of acoustic metamaterial cylindrical curved shells with different multi-resonator parameters. In addition, a periodic graded arrays technique is also utilized to broaden the frequency range of bandgap. Lastly, finite element method (FEM) is used to calculate the band structures and vibration transmission characteristics to verify the accuracy of PWE results. The results disclose that the proposed structure has a good attenuation effect on flexural wave and the periodic graded arrays technique can effectively splice bandgaps of each resonator to enlarge frequency range of the flexural wave mitigation. Hence, the proposed structure can be used to achieve a good attenuation effect on flexural wave in the future.