Bloch-Floquet Theorem Research Articles

An exact spectral formulation for the wave characteristics in an infinite Timoshenko-Ehrenfest beam supported by periodic elastic foundations

Periodic structures have been widely used for vibration reduction and wave control. In the present work, an exact spectral formulation (ESF), being capable to evaluate the dispersion spectra for the bending and shear waves in an infinite Timoshenko-Ehrenfest beam supported by periodic elastic foundations, is proposed, considering the finite length and periodic interval of the distributed elastic supports. First, the frequency-domain displacement function of a unit cell is solved analytically by considering the continuity and equilibrium conditions at the interface between the supported and unsupported parts. Such displacement is formulated as the interpolation of the displacement responses at the end points of the unit cell, and in particular, a compact form is obtained by employing the Hadamard product. Based on this, the spectral matrix for the entire unit cell, involving a total number of 4 degrees of freedom, is derived from the Hamilton’s principle, where the referred matrices are all explicitly given. The dispersion spectra can then be evaluated by solving the eigenvalue problem formulated from the Bloch-Floquet theorem. From a benchmark example, the bandgap characteristics indicated from the proposed ESF have been validated by comparing to the frequency response function of a spectral element model including ten unit cells. Lastly, the distributions of the bandgaps, with respect to the length, the transverse and rotational stiffnesses of the elastic foundation, are investigated based on the defined dimensionless parameters, having potential value to the optimal design of periodic supports for long-span beam structures.

Broadband vibration analysis of the periodically stitched sandwich structure

This work presents a numerical study on the broadband vibration analysis of periodically stitched sandwich structures. Firstly, a finite element model is established for the unit cell, which is capable of taking into account different layers of the sandwich as well as the stitches. The wave propagation properties in different frequency bands are analyzed by applying the Floquet-Bloch theorem to the boundaries of the unit cell. Then, the influences of structural design parameters on broadband dynamic responses are discussed. Simulation results indicate that the dynamic response of the investigated stitched sandwich structure is mainly determined by various far-field propagating waves. Structural dynamic response in the medium and high frequency ranges is significantly suppressed by introducing stitches into the sandwich structure. A more intensive stitch density, larger stitch diameter, and thicker core layer could make the resonance peaks of structural displacement response shift towards lower frequency, which is more obvious in the medium and high frequency bands.

Nonlinearity enhanced wave bandgaps in metamaterial honeycombs embedding spider web-like resonators

Wave propagation in metamaterial honeycombs endowed with periodically distributed nonlinear resonators is addressed. The linear and nonlinear dispersion properties of the metamaterial are investigated. The nonlinear wave propagation equations obtained via a projection method and the Floquet–Bloch theorem are attacked by the method of multiple scales to obtain in closed form the nonlinear manifolds parametrized by the amplitudes, the frequency, and the wave numbers. The effects of the nonlinearity on the frequency bandgaps are thoroughly investigated and the optimization problem of the resonators nonlinearity towards increased bandgap size is tackled to provide a significant practical framework for the design of nonlinear metamaterials.

The propagation behavior of SH surface wave in a semi-infinite magneto-electro-elastic solid with the periodically gold strips

This article theoretically investigates the propagation of shear horizontal (SH) surface wave in the periodically deposited gold strips separated by vacuum on the magneto-electro-elastic (MEE) substrate. Using field analysis theory based on Bloch-Floquet theorem and electromagnetic open boundary conditions, dispersion equations are derived. The MEE composite is comprised of 80% PZT4 and 20% CoFe2O4. The dispersion, wave mode shape, electric and magnetic potentials are being modelled and computed for the SH wave on the MEE substrate. It is observed that the bandgap occurs at resonance when the wavelength of the SH wave matches with periodicity of the grating. The width of the bandgap can be increased or decreased by increasing the strip height to periodicity ratio. It is found that the gold grating on the surface of MEE substrate can traps the SH wave by slowing down the wave when the wave number is less or equal to wave number at the resonance condition. It is observed that above the resonance condition, SH wave shows supersonic behavior where the phase velocity of SH wave is greater than the surface wave velocity. The trapping and supersonic behavior of SH wave are more pronounced as the strip height to grating periodicity ratio increases. As the strip height to periodicity ratio increases, the wave trapping is enhanced and the leakage of wave energy in the substrate is reduced. The results are relevant to design the surface acoustic wave devices.

A metaporoelastic structure that overcomes the sound insulation weaknesses of single and double panel partitions

Single and double panel partitions are well known for their sound transmission loss weaknesses in the vicinity of their structural resonances: the first mode of the finite size panel and the mass-spring-mass resonance of the double panel. In this paper, we investigate the use of a poroelastic based metamaterial to address this problem. The metamaterial is a network of poroelastic beams, the first bending mode of which is tuned at the targeted panel resonance. The porous nature of the resonators allows taking advantage of its dissipative properties stemming from the interaction between solid and fluid phases. For the double panel application, where a porous layer is generally present, the effectiveness of the poroelastic metamaterial relies on the design of its shape and mounting conditions. First, the sound transmission loss of a finite size single panel is studied experimentally in a rectangular duct under normal incidence. It is shown that the poroelastic metamaterial increases the transmission loss up to 12 dB at the first mode of the panel. Using resonators tuned at different frequencies makes it possible to spread this gain. The attenuation mechanisms are analyzed by simulation using a finite element model. Secondly, the double panel partition is investigated with a periodic finite element model using the Floquet-Bloch theorem. Finally, the impact of several parameters like airflow resistivity and filling fraction on the efficiency of the metaporoelastic structure is illustrated considering a diffuse field excitation.

Optimal resonator damping for wave propagation control in mechanical metamaterials

A novel approach to the optimal damping of linearly damped resonators embedded in metamaterial systems is proposed with the aim of minimizing the metamaterials’ response when the external excitation frequency lies within one of the bandgaps. The equation governing wave propagation in the metamaterial system is obtained via a Galerkin projection combined with the quasi-periodicity ansatz of the Floquet–Bloch theorem. It is shown that an optimality criterion can be obtained for the resonator damping for any excitation frequency by extending the Den Hartog theory of fixed points in the frequency response functions. A numerical example of a honeycomb metamaterial is discussed to show how the proposed method works in a practical application. A full numerical optimization is carried out to study the quality factor of the metamaterials’ response with respect to the resonator damping ratio while proving its effectiveness.

Complex dispersion relations and evanescent waves in periodic magneto-electro curved phononic crystal plates

This paper presents a novel semi-analytical method to consider complex dispersion relations and evanescent wave characteristics in periodic magneto-electro-electric curved phononic crystal plates, based on the plate theory and Bloch-Floquet theorem. Generalized elastodynamic wave equation considering the magneto-electric effects in cylindrical coordinate system is established and discretized with the curved higher-order spectral elements. Then, the proposed approach is validated by the analysis of accuracy and convergence. The influence of magneto-electro coupling effect on dispersion relations are discussed. Significant parameters of the periodic curved phononic crystal are studied systematically, which fundamentally determines the tunability of complex dispersion relations of various guided wave modes including propagative wave modes, purely evanescent wave modes, evanescent edge wave modes, and complex wave modes. It is witnessed that propagative wave modes lead the bandgap to the emergence, variation, and closure with the change of significant parameters. And evanescent wave modes dominate two individual patterns of bandgaps. When the radius angle or curvature changes, the polarization of bandgaps appears. Furthermore, complex wave modes are generated with the increase of curvature resulting in strengthening the attenuation of propagative wave modes. The variation of lattice ratio can separate the intersection point of two propagative wave modes, which induces the opening of two bandgaps and the attenuation of evanescent wave modes. The results of this work provides may benefit the multi-functional development and application of smart materials and structures.

Wave propagation behaviors of a low-symmetry reentrant chiral structure with mass inclusion in a single material

This work proposed a low symmetry characteristics metamaterial in a single material. The structural configuration of the metamaterial is composed of the reentrant structure, chiral structure and mass inclusion. The wave propagation characteristics are investigated in terms of the Bloch-Floquet wave propagation theorem. Firstly, the band diagram of the metamaterial is analyzed. The wave behaviors of the structure are changed due to the low symmetry characteristics. The irreducible Brillouin zone (IBZ) of the proposed structure is changed. The band diagram in the whole IBZ is analyzed to get the accurate distributions of the bandgaps. Some extremum values of the dispersion surface are deviated from the boundary of the IBZ. The formation mechanisms of bandgaps are illustrated based on the analysis of the vibration mode shapes. Then, the phase velocity and group velocity of the elastic wave are calculated, showing the strong anisotropic properties. Moreover, the effects of the geometrical parameters on the bandgaps are investigated. The locations and width of the bandgaps can be tuned for by the rational design. The existence of bandgaps of the metamaterial are validated by a finite element model. Finally, a metamaterial beam sample with seven cells is made and experiment test are conducted for the vibration attenuation performance and error analysis. The proposed structure can be used for vibration attenuation in the fields of underwater vehicles.

Coupled flexural and torsional vibration attenuation with locally resonant metamaterials

A metamaterial beam with coupled bending and torsional resonators for broadband flexural–torsional vibration suppression is studied. The main objective is to design and analyse the metamaterial beam for simultaneous attenuation of flexural and torsional vibration waves using FEM and the Bloch-Floquet theorem. The bandgaps of the single unit cell with an infinite periodic structure assumption are studied using Bloch-Floquet reduction with a commercial multi-physics program. Initially, the beam with a bending resonator is examined, and a high-frequency broadband vibration frequency bandgap above 10 kHz is observed. With the addition of a torsional resonator, the existing bending bandgaps are widened, and further new bandgaps are formed. Transmissibility analysis of a metamaterial beam with the above unit cells is performed using the finite element method. The FRFs show similar bandgaps observed in the dispersion curves, thereby validating unit cell analysis. It is observed that coupled bending and torsional resonators create the opportunity to widen and introduce new bandgaps compared to uncoupled resonators. These metamaterial beams with bending and coupled resonators can aid in vibration isolation and frequency filtering of structures subjected to bending and torsional impact loads.

Flexural waves analysis and enhancement of bandgap properties of a periodic track structure

Periodic structures possess frequency bandgaps wherein the waves cannot pass through. Here, the propagation behaviour of vertical and lateral flexural waves and its control in a railway track supported on periodic sleeper blocks connected using fasteners is investigated. The dispersion relationships for two kinds of waves are derived through Floquet–Bloch theorem, and the ensuing band structures are validated from finite element (FE) models. The results demonstrate that a Bragg and a locally resonant (LR) bandgap evolve in the track for both types of waves in the examined frequency range. However, the bandwidth of these bandgaps is found to be very small. Thus, waves can freely propagate in the track for a large frequency range, causing vibration and noise. Subsequently, the dependence of transmission properties of waves on the number of unit cells is studied. It is observed that the attenuation in the bandgap is significantly improved on increasing the number of unit cells. Further, to tune the bandgap properties, a single-degree-of-freedom resonator (SDoF) is used in the middle of each unit cell of the track. Afterwards, the parametric influence of resonator properties, that is, mass, stiffness and damping, on bandgaps is investigated in depth. Moreover, the phenomenon of bandgaps coupling is demonstrated when the resonator is tuned near the Bragg bandgap. The results provided herein are promising to realize the characteristics of flexural waves and to design resonators for track structures.

Sound transmission of active elastic wave metamaterial with double locally resonant substructures surrounded by external mean flow

The elastic wave metamaterial with active feedback control is presented to discuss its dynamic effective parameter and sound transmission properties. Both studies are initiated from the local resonance with the active feedback control and dynamic structural response. The acoustic-structure coupling of the metamaterial with resonators and orthogonal stiffeners are analyzed with effects of the external mean flow. Different analytical methods are used to derive the effective mass density and sound transmission loss (STL) of the elastic wave metamaterial. The effective density is obtained by the dynamic effective medium method firstly. And the second one applies the spatial harmonic expansion method with the principle of virtual work for the STL based on the Bloch–Floquet theorem and Poisson summations. The effective mass density shows frequency-dependent properties and appears positive and negative values in a certain frequency region simultaneously. This research also illustrates both characteristics can be tuned by the active feedback control with the acceleration and displacement feedback control actions. Numerical results of coupled structures show that the changes of transmission characteristics in the certain frequency region are consistent with the effective mass density properties of uncoupled unit cell structure. In addition, influences of the lattice constants, incident angles and Mach number on the STL are illustrated and discussed.

Bandgap characteristics of the two-dimensional missing rib lattice structure

In this paper, the bandgap characteristics of a missing rib lattice structure composed of beam elements are investigated by using the Floquet-Bloch theorem. The tuning of the width and position of the bandgap is achieved by changing the local structural parameters, i.e., the rotation angle, the short beam length, and the beam thickness. In order to expand the regulation of the bandgap, the influence of the material parameters of the crossed long beams inside the structure on the bandgap is analyzed. The results show that the mass density and stiffness of the structure have significant effects on the bandgap, while Poisson’s ratio has no effect on the bandgap. By analyzing the first ten bands of the reference unit cell, it can be found that the missing rib lattice structure generates multiple local resonance bandgaps for vibration reduction, and these bandgap widths are wider. The modal analysis reveals that the formation of the bandgap is due to the dipole resonance of the lattice structure, and this dipole resonance originates from the coupling of the bending deformation of the beam elements. In the band structure, the vibrational mode of the 9th band with a negative slope corresponds to a rotational resonance, which is different from that with the conventional negative slope formed by the coupling of two resonance modes. This study can provide a theoretical reference for the design of simple and lightweight elastic metamaterials, as well as for the regulation of bandgaps and the suppression of elastic waves.

Dynamic Analysis and Design of Metamaterial Plates With Crossed Acoustic Black Holes for Vibration Control

Abstract Acoustic black holes (ABHs) have shown great potential in vibration and noise control. Merging the ABH effect and the metamaterial can be a more efficient approach for vibration control. The aim of this article is to study the dynamics of a metamaterial plate with crossed acoustic black holes. The band gap properties of the infinite structure and the influence of the design variables are investigated by using the finite element method and the Floquet–Bloch theorem. The vibration transmission and frequency response functions of the finite structure are presented to reveal the vibration attenuation mechanism. The effect of elastic boundary conditions on the vibration properties of the metamaterial plate is also studied. Numerical results demonstrate that the vibration is remarkably weakened due to the band gap and local modes induced by the ABH effect. Then, experimental validation is given by using 3D printing techniques. Finally, we study the multi-objective optimal design problem of the ABH plate to reduce the vibration amplitude and the structural mass simultaneously. Optimization results provide more options for the trade-off design of metamaterial plates between the lightweight design and vibration suppression capability.

Flexural wave propagation in periodic Micropolar-Cosserat panels: Spectral Element Formulation

In this research, a perspective of the dispersion relation of flexural waves on periodic structures is proposed. The rotational theory of micro-elements is stem to explore the favorable impact of size effect on structures through the localization of the strain energy. The panels are modelled as a one-dimensional (1-D) Micropolar–Cosserat (MC) continuum to explore the effect of length-scale or micro-rotational waves of solids. The spectral element formulation of a panel and the transfer matrix of the unit cell is presented within the framework of the state-space method. The propagation constant in the eigenvalue domain is established based on the Bloch–Floquet theorem to account for the periodicity of the panel. The MC theory in analyzing the band-gap (stop-band) and pass-band characteristics of the unit cell panels is validated by a commercial finite element software package-COMSOL Multiphysics. The mode shape of unit cell for the start and end point of band-gap is further analyzed within the realms of finite element method (FEM) analysis. The appropriate limiting condition reverts MC spectral element into Timoshenko (TM) beam model. The comparison of Timoshenko beam model with the spectral element of 1-D plane-stress (PS) analysis is also investigated. It is observed that the shear wave of the MC model is coupled with a micro-rotational wave of micro-elements, unlike the classical continuum framework. The response of the finite structure subjected to various frequencies is presented too. Presence of vibration attenuation in the specified band-gap (BG) is justified through the elemental dynamic stiffness (DS) matrix of unit cells. This reduced the level of vibration at the specific frequency range within the attenuation band or disintegrates it into two bands. The attenuation bandwidth and its location is significantly affected by the modulation of both geometry and material properties of the beam. Investigation of flexural wave propagation based on the proposed 1-D MC beam theory shows good agreement with the two-dimensional (2-D) plane-stress FEM analysis.

Energy dissipation enhancement of flexural metamaterial beams with inerter and rotational deformation

Elastic metamaterials with locally resonant components exhibit unique band-gap behavior that can be applied to control wave propagation in flexural beams. Previous studies mainly emphasized the utilization of transversal deformations of flexural beams. A novel metamaterial configuration is presented in this work; in particular, to obtain a wide band gap in low-frequency ranges with reduced expenses in associated mass, inerter-based resonators and cantilevered deformations caused by the rotation of flexural beams are combined. The study is performed by using an analytical approach based on the finite element (FE) method and the Floquet-Bloch theorem. Both the wavenumber-frequency and the damping ratio-frequency relationships and relevant band structures are set. These two types of band structures are further considered for the investigation of the effects of various system parameters on the band-gap behavior and wave attenuation performance in the whole frequency range. Finally, the effectiveness of the proposed concept is validated by means of numerical examples.

Bandgap manipulation of single and multi-frequency smart metastructures with random impedance disorder

Metamaterials and metastructures usually exhibit some form of periodicity that is broken due to randomness in the media, affecting the vibration and elastic wave propagation performance. Disorder in the periodicity caused by the variability in properties or geometry may lead to interesting physical phenomena such as trapping and scattering waves, wave reflection, and energy localisation. While the randomness may be attributed to manufacturing irregularities and quantifying its effect is crucial for ensuring adequate performance of a range of smart systems, and these effects can also be exploited for manipulating the wave properties. Here we investigate a smart metastructure in the form of a beam integrated with piezoelectric transducers coupled to a resonant shunt circuit. The piezoelectric shunt in a periodical arrangement can induce locally resonant bandgaps that can be employed in wave (and vibration) manipulation and control. This paper quantifies the uncertainty associated with electrical circuit components that affect the circuit impedance influencing wave and vibration control features. The smart unimorph meta-beam is formulated by spectral element method based on the Hamiltonian principle, and dispersion curves are estimated using the spectral transfer matrix method that relies on the Bloch-Floquet theorem. The first and second statistical moments are obtained from the stochastic simulations ( realizations) that could be realized efficiently due to the current development of spectral element approach for analysing the disordered system. Further experimental validation and exploration of shunt circuit confirms the need to include the stochastic information on the numerical model used to design the metastructure. Numerical results of the unimorph meta-beam with single and multi-frequency shunt configuration show that the bandgap behaviour is sensitive to the random disorder associated with circuit impedance parameters. Such disorder can be exploited for enhanced functionalities of the meta-beam based on optimal RL shunt circuits for controlling structural vibration response along with wave propagation and attenuation.

Enhanced attenuation band in active 2D mass-in-mass metamaterial using static output feedback

Active feedback control is implemented between the inner and outer masses of a 2D mass-in-mass metamaterial to widen the attenuation bandwidth by reducing the displacement of the outer mass of each unit cell. The optimal static output feedback control is applied to a single representative unit. The gain is computed from the analysis of a single unit cell and further used along with the Bloch–Floquet theorem for obtaining the dispersion relationship for an infinite 2D discrete square lattice metamaterial. The proposed model is validated by comparing the results against the finite element simulation in ANSYS. The control gain couples the displacements and velocities, which generates the damping phenomenon even in an undamped system. The controller design requires feedback of only a few measured output instead of all the state variables, and offline computation of the feedback gains reduces the time delay. Effective mass analysis and energy dissipation mechanism are performed which sheds light on the physics underlying the attenuation phenomena. An ultra-wide uninterrupted attenuation band can be perceived as the negative effective mass region significantly extends beyond the standard limit by employing the active feedback control. Additionally, the attenuation bandwidth exists in the low frequency due to the damping induced from the active control.

Enhancing the sound transmission loss in double-leaf partitions with lateral local resonators substructure

We report theoretically on the sound transmission loss performance through a periodic double plate acoustic metamaterial made of lateral local resonance (LLR) substructure. The unit cell of the substructure consists of a four link mechanism, two lateral resonators, and a vertical spring. The combination of space harmonic expansion and Bloch-Floquet theorem are used to analyze this present study. Computed results show that high sound transmission loss (STL) up to 60 dB at 62 Hz is reached with the metamaterial plate while the mass spring mass resonant is observed for the conventional periodic double plate at the same frequency. The introduction of a negative stiffness spring causes an increased STL at low frequency. The potential of lateral resonant metamaterials to improve the sound transmission loss (STL) in the frequency region around the mass-spring-mass resonance for periodic double panel partitions is demonstrated.

Two-Dimensional In-Plane Elastic Waves in Curved-Tapered Square Lattice Frame Structure

Abstract This work proposes a unique configuration of a two-dimensional metamaterial lattice grid comprising curved and tapered beams. The propagation of elastic waves in the structure is analyzed using the dynamic stiffness matrix (DSM) approach and the Floquet–Bloch theorem. The DSM for the unit cell is formulated under the extensional theory of curved beam, considering the effects of shear and rotary inertia. The study considers two types of variable rectangular cross sections, viz. single taper and double taper along the length of the beam. Further, the effect of curvature and taper on the wave propagation is analyzed through the band diagram along the irreducible Brillouin zone. It is shown that a complete band gap, i.e., attenuation band in all the directions of wave propagation, in a homogeneous structure can be tailored with a suitable combination of curvature and taper. Generation of the complete bandgap is hinged upon the coupling of the axial and transverse components of the lattice grid. This coupling emerges due to the presence of the curvature and is further enhanced due to tapering. The double taper cross section is shown to have wider attenuation characteristics than single taper cross sections. Specifically, 83.36% and 63% normalized complete bandwidth is achieved for the double and single taper cross section for a homogeneous metamaterial, respectively. Additional characteristics of the proposed metamaterial in the time and frequency domain of the finite structure, vibration attenuation, wave localization in the equivalent finite structure are also studied.

Extraordinary optical characteristics of one-dimensional double anti-PT-symmetric ring optical waveguide networks

In this paper, a unique one-dimensional double anti-PT-symmetric ring optical waveguide network (1D DAPTSROWN) is designed whose refractive index satisfies n(x)=−n∗(−x). The extraordinary optical characteristics of the network are studied by means of the generalized Floquet–Bloch theorem, three-material network equation and generalized eigenfunction method. Due to the physical difference between the anti-PT-symmetric and PT-symmetric optical waveguide networks, the previous method of the PT-symmetric optical waveguide network is not applicable to the anti-PT-symmetric optical waveguide network designed in this paper. Thus, based on the previous method and the network equation, we propose a connection function to identify the extremum mode points in the anti-PT-symmetric optical waveguide network. At the extremum mode point, the optical waveguide network can produce extraordinary ultra-strong transmission, ultra-strong reflection, and ultra-strong photonic localizations. The corresponding calculated values in this paper arrives at 1028, which is much larger than 1012 (the best result reported previously). This shows that the anti-PT-symmetric network may have potential in applications in designing optical filters, optical amplifiers and so on.