Lamb Wave Band Gaps Research Articles

Precise and target-oriented control of the low-frequency Lamb wave bandgaps

Precise and target-oriented control of acoustic/elastic wave bandgaps in phononic crystals or acoustic metamaterials according to particular requirements is of both theoretical and practical importance. A very challenging and also interesting task is to tune the widths and the overall positions of the bandgaps of interest independently and arbitrarily. This work presents a simple and smart acoustic metamaterial structure, which can achieve the above mentioned goal by considering the symmetric Lamb waves. The proposed acoustic metamaterial structure consists of a homogenous piezoelectric plate which is connected to the LC circuits through an array of periodic surface electrodes. The control of the resonance bandgaps (RBGs), which are generated by the coupled resonance of the piezoelectric plate with the external LC circuits, is the focus of this study. To solve the problem at hand, we develop a two-dimensional (2D) spectral element method (SEM) considering the full electro-mechanical coupling, and the high tunability of the RBGs is demonstrated numerically in the low-frequency range. Especially, we find that the upper/lower boundary of a RGB can be intentionally fixed at any specified frequency while changing its’ bandwidth. To explore the corresponding physical phenomena and reveal the regulation mechanism, a simplified theory is developed based on the assumption of thin piezoelectric plates. Several simple analytical formulas are established, which are essential to the precise control of the RBGs for both the thin and thick piezoelectric plates. A numerical example is given to demonstrate the detailed procedure of precisely tuning a RBG in a targeted frequency range, which is also verified by the transmission spectrum for the corresponding finite piezoelectric acoustic metamaterial plate.

Gap evolution of Lamb wave propagation in magneto-elastic phononic plates with pillars and holes by modulating magnetic field and stress loadings

Considering the nonlinear coupling behavior of magnetostrictive material, the modulation of Lamb wave bandgaps in magneto-elastic phononic plates composed of Terfenol-D pillars on a silicon matrix is investigated by the finite element method. By the introduction of holes, two schemes, i.e., the pillars only case for scheme-I and the trampoline (pillars and holes) case for scheme-II, are considered for exploring the effect of magnetostriction and trampoline on band structures. Numerical results show that the edges of bandgaps shift toward higher frequencies and the relative bandwidth enlarges as the magnetic field increases. The greater the compressive pre-stress applied, the greater the magnetic field at the open or closed points of the bandgap required. Compared to scheme-I, we find that the existence of holes for scheme-II can cause the closing of the higher branches’ bandgaps and the generation of a new bandgap, and larger relative bandwidth of the bandgap and wider range of the required magnetic field can be observed due to the trampoline effect. Meanwhile, the height of the pillar is a key parameter for generating or vanishing bandgaps. According to the displacement distribution of eigenmodes, it can be seen that the opening or closing of the bandgap is controlled by the coupling between Lamb modes of the plate and resonant modes of the pillars, which is induced by the combined effect of trampoline, magnetic field, and pre-stress as well as geometry parameters. These results give guidance for active controllability of Lamb wave propagation and intelligent regulation of phononic devices in complex environments.

The Lamb wave bandgap variation of a locally resonant phononic crystal subjected to thermal deformation

A study on dynamical characteristics of a ternary locally resonant phononic crystal (PC) plate (i.e., hard scatterer with soft coating periodically disperse in stiff host matrix) is carried out in this paper. The effect of thermal deformation on the structure stiffness, which plays an important role in the PC’s dynamical characteristics, is considered. Results show that both the start and the stop frequency of bandgap shift to higher range with the thermal deformation. In particular, the characteristics of band structure change suddenly at critical buckling temperature. The effect of thermal deformation could be utilized for tuning of phononic band structures, which can promote their design and further applications.

Tunable Lamb wave band gaps in two-dimensional magnetoelastic phononic crystal slabs by an applied external magnetostatic field

This paper theoretically investigates the band gaps of Lamb mode waves in two-dimensional magnetoelastic phononic crystal slabs by an applied external magnetostatic field. With the assumption of uniformly oriented magnetization, an equivalent piezomagnetic material model is used. The effects of magnetostatic field on phononic crystals are considered carefully in this model. The numerical results indicate that the width of the first band gap is significantly changed by applying the external magnetic field with different amplitude, and the ratio between the maximum and minimum gap widths reaches 228%. Further calculations demonstrate that the orientation of the magnetic field obviously affects the width and location of the first band gap. The contactless tunability of the proposed phononic crystal slabs shows many potential applications of vibration isolation in engineering.

Lamb wave band gaps in one-dimensional radial phononic crystal plates with periodic double-sided corrugations

In this paper, we present the theoretical investigation of Lamb wave propagation in one-dimensional radial phononic crystal (RPC) plates with periodic double-sided corrugations. The dispersion relations, the power transmission spectra, and the displacement fields of the eigenmodes are studied by using the finite element method based on two-dimensional axial symmetry models in cylindrical coordinates. Numerical results show that the proposed RPC plates with periodic double-sided corrugations can yield several band gaps with a variable bandwidth for Lamb waves. The formation mechanism of band gaps in the double-sided RPC plates is attributed to the coupling between the Lamb modes and the in-phase and out-phases resonant eigenmodes of the double-sided corrugations. We investigate the evolution of band gaps in the double-sided RPC plates with the corrugation heights on both sides arranged from an asymmetrical distribution to a symmetrical distribution gradually. Significantly, with the introduction of symmetric double-sided corrugations, the antisymmetric Lamb mode is suppressed by the in-phase resonant eigenmodes of the double-sided corrugations, resulting in the disappearance of the lowest band gap. Furthermore, the effects of the geometrical parameters on the band gaps are further explored numerically.

Extremely low-frequency Lamb wave band gaps in a sandwich phononic crystal thin plate

In this paper, a kind of sandwich phononic crystal (PC) plate with silicon rubber scatterers embedded in polymethyl methacrylate (PMMA) matrix is proposed to demonstrate its low-frequency Lamb wave band gap (BG) characteristics. The dispersion relationship and the displacement vector fields of the basic slab modes and the locally resonant modes are investigated to show the BG formation mechanism. The anti-symmetric Lamb wave BG is further studied due to its important function in reducing vibration. The analysis on the BG characteristics of the PC through changing their geometrical parameters is performed. By optimizing the structure, a sandwich PC plate with a thickness of only 3 mm and a lower boundary (as low as 23.9 Hz) of the first anti-symmetric BG is designed. Finally, sound insulation experiment on a sandwich PC plate with the thickness of only 2.5 mm is conducted, showing satisfactory noise reduction effect in the frequency range of the anti-symmetric Lamb BG. Therefore, this kind of sandwich PC plate has potential applications in controlling vibration and noise in low-frequency ranges.

Lamb wave band gaps in one-dimensional radial phononic crystal slabs

In this paper, we theoretically investigate the band structures of Lamb wave in one-dimensional radial phononic crystal (PC) slabs composed of a series of alternating strips of epoxy and aluminum. The dispersion relations, the power transmission spectra and the displacement fields of the eigenmodes are calculated by using the finite element method based on two-dimensional axial symmetry models in cylindrical coordinates. The axial symmetry model is validated by three-dimensional finite element model in Cartesian coordinates. Numerical results show that the proposed radial PC slabs can yield several complete band gaps with a variable bandwidth exist for elastic waves. Furthermore, the effects of the filling fraction and the slab thickness on the band gaps are further explored numerically. It is worth observing that, with the increase of the filling fraction, both the lower and upper edges of the band gaps are simultaneously shifted to higher frequency, which results from the enhancement interaction between the rigid resonance of the scatterer and the matrix. The slab thickness is the key parameter for the existence and the width of complete band gaps in the radial PC slabs. These properties of Lamb waves in the radial PC plates can potentially be applied to optimize band gaps, generate filters and design acoustic devices in the rotary machines and structures.

Enlargement of the locally resonant Lamb wave band gap of the phononic crystal plate at the deep sub-wavelength scale

In this paper, the numerical investigation of Lamb wave propagation in two-dimensional phononic crystals composed by depositing the heavy cylinder stubs squarely onto both sides of a composite thin plate is presented. A significant enlargement of the relative bandwidth by two orders of magnitude compared with two-dimensional binary locally resonant phononic crystal plates and one order of magnitude compared with both the ternary composite plate and the simple plate with stub structures at deep sub-wavelength scale is obtained and discussed. Based on an efficient finite element method, we show that this band gap enlargement is due to the coupling between the plate mode and ‘double mass-spring’ mode, which leads to the enhancement of the locally resonance mechanism. The calculated elastic wave displacement fields illustrate the physics mechanism of opening the low frequency band gap and the existing of the flat bands. We also found that both the band gap and the flat bands depend on the geometry parameters of the heavy stubs and composite plate significantly.

Lamb wave band gaps in a homogeneous plate with inclined cylindrical dots

We report on the theoretical analysis of locally resonant sonic band gaps (BGs) in a phononic crystal (PC) structure constituted by a square array of inclined cylindrical dots deposited on a thin homogeneous plate. Based on an efficient finite element method (FEM), we show that the PC plate with inclined dots can lower Lamb wave BGs due to a weak localized resonance of the edge of the upper surface of the inclined dots. The BGs can be effectively shifted by changing the value of the incline. The total displacement fields are computed to explain how the inclined dot induces the lowering of the locally resonant BGs. Transmission power spectra (TPS) are calculated to demonstrate the existence of the BGs and it matches well with the band structure. BGs evolution as a function of geometrical parameters of the PC plate with inclined dot is also studied.

Propagation of Lamb waves in one-dimensional radial phononic crystal plates with periodic corrugations

In this paper, we theoretically investigate the propagation characteristics of Lamb waves in one-dimensional radial phononic crystal plates with periodic corrugations. The dispersion relations, the power transmission spectra, and the displacement fields of the eigenmodes are calculated by using the finite element method based on two-dimensional axial symmetry models in cylindrical coordinates. The axial symmetry model is validated by three-dimensional finite element model in rectangular coordinates. The effects of the geometrical parameters on the band gaps are further explored numerically. Numerical results show that several complete band gaps with a variable bandwidth exist for Lamb waves in the proposed structures. The formation mechanism of opening the acoustic band gaps is attributed to the coupling between the Lamb modes and the corrugation mode. The band gaps are significantly dependent upon the geometrical parameters such as the corrugation height, the corrugation width, and the plate thickness. Significantly, as the increase of corrugation height, band width shifts, new band gaps appear, the bands become flat, and the corrugation mode plays a more prominent role in the opening of Lamb wave band gaps. These properties of Lamb waves in the radial phononic crystal plates can potentially be applied to optimize band gaps, generate filters, and design acoustic devices.

Lamb wave band gaps in a homogenous plate with periodic Gaussian surfaces

The Lamb wave band structure of a one-dimensional thin plate with periodic Gaussian surfaces is calculated based on the eigen-mode matching theory (EMMT) method. The finite element method is also employed to calculate the transmission power spectra and the band structure, which are both in good agreement with the results of the EMMT method. In contrast with the straight stubs, the low-frequency band gap of the system with the Gaussian surfaces is wider, when the Gaussian domain's height and area are equal to those of the stub, and the two systems' main plate size and materials are the same. This can be confirmed by the ‘spring–mass' model, which can be used to estimate the resonant frequency. We also investigate the effects of the geometrical parameters on band structure, including the height of the Gaussian domain, the plate thickness and the filling fraction. The results suggest that the band gaps have a close relevance to them.

Lamb wave band gaps in a double-sided phononic plate

In this paper, we report on the theoretical investigation of the propagation characteristics of Lamb wave in a phononic crystal structure constituted by a square array of cylindrical stubs deposited on both sides of a thin homogeneous plate. The dispersion relations, the power transmission spectra, and the displacement fields of the eigenmodes are studied by using the finite-element method. We investigate the evolution of band gaps in the double-sided phononic plate with stub height on both sides arranged from an asymmetrical distribution to a symmetrical distribution gradually. Numerical results show that as the double stubs in a unit cell arranged more symmetrically on both sides, band width shifts, new band gaps appear, and the bands become flat due to localized resonant modes which couple with plate modes. Specially, more band gaps and flat bands can be found in the symmetrical system as a result of local resonances of the stubs which interact in a stronger way with the plate modes. Moreover, the symmetrical double-sided plate exhibits lower and smaller band gap than that of the asymmetrical plate. These propagation properties of elastic or acoustic waves in the double-sided plate can potentially be utilized to generate filters, slow the group velocity, low-frequency sound insulation, and design acoustic sensors.

Lamb wave band gaps in a homogenous plate with periodic tapered surface

In this paper, we present the numerical investigation of Lamb wave propagation in a homogenous plate with periodic tapered surface, which gradually increases the width from the lower base to the upper base. The dispersion relations, the power transmission spectra, and the displacement fields of the eigenmodes are studied by using the finite-element method. We investigate the effects of the geometrical parameters (including the ratio of the lower base width to the upper base width, and the ratio of the upper base width, the thickness of the tapered surface, and the thickness of the homogenous plate, respectively, to the lower base width) on the band gaps. Numerical results show that the band gaps can be effectively shifted by changing the geometrical parameters. Especially, the width of the first band gap changes approximately linearly by changing the ratio of the upper base width to the lower base width and in return. The transmission bands of the structure with the tapered surface are more flat than those of the structure with the stubbed surface. Moreover, the proposed homogenous plate with periodic tapered surface exhibits lower and smaller band gap than that of the homogenous plate with periodic stubbed surface due to weak localized resonance of the tapered surface with the upper base wider than the lower base. These properties of elastic or acoustic waves can potentially be utilized to tune band gaps, slow the group velocity, generate filters, and design acoustic sensors.

Thermal tuning of Lamb wave band structure in a two-dimensional phononic crystal plate

Using supercell-plane-wave-expansion method, we have made a theoretical study of the temperature effects on the band structure of Lamb wave in a two-dimensional phononic crystal plate. The numerical results show that the Lamb wave bandgap is very sensitive to temperature. For the case of Ba0.7Sr0.3TiO3/epoxy phononic crystal, the width of bandgap becomes narrower in a hexagonal lattice system as the temperature increases, but it becomes wider in a phononic crystal plate with square lattice. On the other hand, for the case of air/Ba0.7Sr0.3TiO3 plate with square lattice, wider and higher complete bandgap can be created in a higher temperature. These results are quite different from that in the infinite phononic crystal and one-dimensional phononic crystal plate.

Lamb wave band gaps in locally resonant phononic crystal strip waveguides

Using finite element method, we have made a theoretically study of the band structure of Lamb wave in a locally resonant phononic crystal strip waveguide with periodic soft rubber attached on the two sides of epoxy main plate. The numerical results show that the Lamb wave band gap based on local resonant mechanism can be opened up in the stub strip waveguides, and the width of the local resonant band gap is narrower than that based on the Bragg scattering mechanism. The results also show that the stub shape and width have influence on the frequency and width of the Lamb wave band gap.

Propagation characteristics of the laser-generated lower-order lamb waves in one-dimensional periodic thin plates

Using the finite element method (FEM), we study the thermoelastic generation and the transmitted power spectra (TPS) of Laser-generated lower-order Lamb wave mode in an one-dimensional thin aluminum plate engraved by a periodic rectangular grating. The calculation verified that the equally good Lamb waves band gaps can also exist in an one-component periodic structure and the width and location of the band gaps can be regulated by the parameters of the depth of the grooves (h), the ratio of the lattice period to the plate thickness (D/L) and the ratio of the width of the grooves to the lattice period (d/D). Specially, we suggest that Lamb wave band gaps can be substantially enlarged by using multiple periodic systems which consist of several pieces of periodic structure with different D/L.

Low-frequency band gaps in one-dimensional thin phononic crystal plate with periodic stubbed surface

Using supercell plane wave expansion method, the Lamb wave band structure of one-dimensional thin plate with periodic stubs is investigated. The numerical results show that flat bands will appear and band gap can exist in a low-frequency domain. The position of the flat bands and width of the low-frequency Lamb wave band gap can be tuned by the stub height, plate thickness and filling fraction. The band gap is obtained by opening the folding points of the same plate modes not the crossing point of different plate mode when the stub height is small.

Temperature effects on the band gaps of Lamb waves in a one-dimensional phononic-crystal plate (L)

This study investigates the temperature-tuned band gaps of Lamb waves in a one-dimensional phononic-crystal plate, which is formed by alternating strips of ferroelectric ceramic Ba(0.7)Sr(0.3)TiO(3) and epoxy. The sensitive and continuous temperature-tunability of Lamb wave band gaps is demonstrated using the analyses of the band structures and the transmission spectra. The width and position of Lamb wave band gaps shift prominently with variation of temperature in the range of 26 °C-50 °C. For example, the width of the second band gap increases from 0.066 to 0.111 MHz as the temperature is increased from 26 °C to 50 °C. The strong shift promises that the structure could be suitable for temperature-tuned multi-frequency Lamb wave filters.

Engineering band gap forming using the combination of periodic and Fibonacci quasiperiodic one-dimensional composite thin plates

We suggest that Lamb wave band gap in one-dimensional phononic crystals can be substantially enlarged by using the combination of periodic and Fibonacci quasiperiodic composite thin plates. A narrow band also can be obtained by using Fibonacci quasiperiodic or the combined composite thin plates. This is possible because each substructure shows a different band gap which is different from those in the phononic crystals and photonic crystals of the bulk waves. This effect has been demonstrated by using the finite element method.

Calculations of Lamb wave band gaps and dispersions for piezoelectric phononic plates using mindlin's theory-based plane wave expansion method

Based on Mindlin's piezoelectric plate theory and the plane wave expansion method, a formulation is proposed to study the frequency band gaps and dispersion relations of the lower-order Lamb waves in two-dimensional piezoelectric phononic plates. The method is applied to analyze the phononic plates composed of solid-solid and airsolid constituents with square and triangular lattices, respectively. Factors that influence the opening and width of the complete Lamb wave gaps are identified and discussed. For solid/solid phononic plates, it is suggested that the filling material be chosen with larger mass density, proper stiffness, and weak anisotropic factor embedded in a soft matrix in order to obtain wider complete band gaps of the lower-order Lamb waves. By comparing to the calculated results without considering the piezoelectricity, the influences of piezoelectric effect on Lamb waves are analyzed as well. On the other hand, for air/solid phononic plates, a background material itself with proper anisotropy and a high filling fraction of air may favor the opening of the complete Lamb wave gaps.