Wave Mode Shapes Research Articles

Shear horizontal wave propagation on a piezoelectric substrate with periodic gradient local resonant metasurfaces

In this paper, analytical solutions for shear horizontal (SH) waves propagating in a piezoelectric (PE) substrate with periodic gradient mass oscillators are obtained. Both wave-mode method and spectral method are used to obtain the dispersion equation of SH waves. The influences of mass-spring oscillators on dispersion curves and the wave mode shapes are discussed via numerical examples. It reveals that the numerical results obtained by wave-mode method and spectral method are well matched. In addition, the mass-spring oscillators have evident effects on the propagation characteristic of SH waves, and the effects are closely related with the frequency. Moreover, the widths of the local resonant and Bragg resonant bandgaps become evident wider when oscillator's natural resonance frequency is close to the Bragg scattering frequency. In particular, based on our proposed structure, we can design a sensor with advantages such as filtering and only sensitive to specific frequencies, amplify lossless signals and multi frequency sensitivity.

A new Stroh formalism for gradient electro-mechanics with applications to Lamb waves in piezoelectric and flexoelectric coupled plates

In this paper, a new Stroh formalism for gradient electro-mechanics is derived for the first time, which is both mathematically concise and numerically powerful, applicable to generally coupled anisotropic material systems. Based on this new formalism, the complicated Lamb wave in flexoelectric and piezoelectric plates is investigated. The dispersion equation is obtained by solving the eigenvalue problem along with the unconditionally stable dual-variable and position method. From the obtained dispersion equation, the dispersion curves and mode shapes of the Lamb wave are calculated by the 1D form of the multidimensional moduli ratio convergence method. Two important and interesting features are observed from our analysis: One is the difference in the mode shape symmetry between the piezoelectric and flexoelectric cases, and the other is the size-dependent property of the flexoelectric effect as observed by nondimensionalization. These features are further illustrated by comparing the dispersion curves and wave-mode shapes among the three different material models (purely piezoelectric, purely flexoelectric, and flexoelectric and piezoelectric coupled). The newly derived Stroh formalism offers a robust, concise, and unified approach for dealing with strain gradient electro-mechanic materials with crystal systems of general anisotropy. The present work also explains the physical mechanism of symmetry breaking observed, as induced by flexoelectric coupling in piezoelectric materials.

GHz surface waves in Al/LiTaO3/Si composite: Effect of the Drude electrode on dispersion, attenuation and mode shapes

In fifth generation (5G) mobile communication, radio frequency is in the super high frequency range (3–30 GHz). Thus, different from Ohm law, electrode loss at GHz frequency becomes frequency-dependent. As such, many properties, especially attenuation of piezoelectric acoustic devices, should be reevaluated. In this paper, the attenuation of GHz surface wave propagating in composite structure made of metal electrode, piezoelectric film and elastic half space is investigated based on the loss model of Drude electrode. Specifically, the Al/LiTaO3/Si composite is considered where Rayleigh mode and SH mode are fully coupled. Based on the Stroh-type formalism, the dispersion equations of the composite with the Drude electrode are derived, and compared with the traditional perfect conductor electrode which is under the short circuit boundary condition. The phase velocity, attenuation and wave mode shapes are obtained successfully by the multidimensional moduli ratio convergence method. Numerical results are presented to reveal some interesting intrinsic features, which include two analytical upper bounds of phase velocity, nonexistent validation of intersection points in dispersion curves, and relations among veering, attenuation jumps and wave-mode conversions. These features and relations provide an in-depth insight for GHz wave motion properties in piezoelectric composites.

Traveling wave modes of flexible structures with distributed gyros

With the development of space structures, flexible structures are widely studied. To achieve attitude and vibration control of flexible structures, flywheels or control moment gyros can be introduced. However, the critical challenge in studying gyro flexible structures is establishing a systematic framework to manage its physical properties and calculate the control moment precisely. This study aims to construct a theoretical framework for a two-dimensional flexible structure with distributed gyros using classical methods and explore the contribution of the gyro moment on the dynamic mode shapes. Both eigenvalue and traveling wave mode shape analysis are applied to investigate the dynamical behaviors of the system. It is found that dynamic modes with nodal lines showing whirling phenomena have arisen instead of the traditional modes featured with static nodal lines triggered by the gyro moment, and the dynamic mode shapes can be displayed by the direction and frequency of the mode nodal lines motion. The discovery and research of gyro flexible structure traveling wave modes can provide a solid base for the design of large space structures, improve their stability, and avoid harmful vibrations.

Bolt Axial Stress Measurement Based on the Dual-Mode Electromagnetic Acoustic Transducer

To further simplify the bolt stress measurement process, the feasibility of using a dual-mode EMAT that excites both longitudinal and shear waves is investigated to achieve rapid and accurate measurement. The propagation characteristics of cylindrical guided waves in a rod are studied by a Semi-analytical Finite Element model. Through the combination of the theoretical formula and the finite element simulation, the non-uniform distribution of axial stress is directly used to calculate the time of flight of cylindrical guided waves without building any approximate model. The effect of stress distribution on guided wave modes with different energy concentration regions is analyzed. The propagation characteristics of guided waves and the theoretical values of group velocity of these two modes are studied by cylindrical guided wave dispersion curve and mode shape analysis. Then, by combining the static stress distribution inside the bolt with the theoretical formula, the influence of the different distributions of guided wave energy in the cross section of the two modes on the velocity ratio was analyzed. Finally, the feasibility of the theory is verified by the experiment. The result of the experiment in the aluminum bolt verified the effectiveness of the dual-mode EMAT on stress measurement and the novel ultrasonic testing models proposed in this paper.

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.

Rayleigh wave in layered piezoelectric semiconductor with consideration of PN junction effects

In this paper, analytical solutions for Rayleigh wave in a transversely isotropic piezoelectric semiconductor (PSC) substrate with a covering layer of different type doping are obtained. The PN junction (called also the depletion layer), which is created by the diffusion and drifting due to the different doping type of the substrate and the covering layer, can be considered as an electrically gradient layer between the covering layer and the substrate. Making use of the state transfer equation method, the transfer matrix of the state vector of the depletion layer is derived mathematically. Based on the numerical calculations, we investigate the physical influences of the covering layer, electric boundary conditions and doping concentrations on the dispersion and attenuation, and the effects of PN junction on the wave-mode shapes. Our results show that the semiconductor will slow the Rayleigh waves and induce the attenuation. Both the phase velocity and attenuation are dependent upon the frequency. The existence of PN junction has less influences on the dispersion and attenuation but relatively large influences on the wave-mode shapes, especially, on the electric potential, carriers concentration, electric displacement, and the current density of carriers.

A new model for acoustic attenuation of GHz waveguide induced by metal electrodes in piezoelectric composites

• A new model based on electromagnetic (or optic) property of metal is established. • 3D dispersion curves with energy dissipation are investigated by a novel root-searching method. • A rapid conversion phenomenon in wave mode shapes is identified. • Two new features which significantly change the acoustic attenuation are observed. • Besides the electromagnetic property , mechanical property of electrode also impacts the attenuation in GHz. Acoustic attenuation in piezoelectric composites is an important factor and it can be induced by various sources, including viscoelasticity and semi-conductivity. Since the radio frequency of fifth generation (5G) mobile communication technology has achieved at GHz, the attenuation of the piezoelectric acoustic devices operating at GHz frequency must be addressed. Unfortunately, when frequency is high, the electrode cannot be treated as an electric resistance that follows the traditional Ohm's law. Therefore, in this paper, the acoustic attenuation induced by electrodes, which has been seldom studied in the past, is investigated. A novel multi-disciplinary model is presented which combines the piezoelectric theory and the electromagnetic (or optic) property of metal, instead of the zero-thickness surface electrode model and Ohm's law with a fixed conductivity. Based on this model, the dispersion equations for the composite structure are derived in the case of elastic metal electrode with realistic optical properties, as compared with the ideal short circuit boundary condition case. Three-dimensional (3D) dispersion curves of these two cases are further calculated by two forms of the novel root-searching method introduced recently, accompanying with the wave mode shapes on certain branches. Based on these results, two important features of the wave motion, which significantly change GHz acoustic attenuations, are identified, with their underlying mechanisms. These new features could provide a reference in safe and better design of piezoelectric acoustic devices.

Determination of Modes in the Ice–Water Bi-layer Waveguide

The seismic method on ice provides rich information and a feasible technical way for acoustic study in ice-covered water. This paper presents a method to determine the modes in ice–water bi-layer waveguides. We achieve the purpose by deducing the scaled boundary formulation for solid and fluid layers then coupling them with the coupling matrix. The wave numbers and mode shapes are compared with the results of the finite element method. It shows that the configuration for ice–water bi-layer waveguide is of high precision. In underwater acoustic applications, the depth of water is comparatively large relative to ice thickness, which addresses the additional difficulty of accuracy in high frequency. The study on convergence is carried out, and approximate formulas are addressed based on the calculated results, giving a quick insight into piratical application.

Shear horizontal wave in a piezoelectric semiconductor substrate covered with a metal layer with consideration of Schottky junction effects

In this paper, the shear horizontal wave in a transversely isotropic piezoelectric semiconductor substrate covered with a metal layer are obtained. The Schottky junction, which is created by the n-type piezoelectric semiconductor and the metal with a lower Fermi level, is considered as an electrically gradient layer between the layer and the substrate. Making use of the state transfer equation method, the transfer matrix of the state vector of the space charge region is derived mathematically. Basing on the numerical calculations, we investigate the physical influences of the doping density, Schottky junction and covered layer thickness on the dispersion and attenuation and the effects of Schottky junction on the wave mode shapes. Our results show that the Schottky junction has negligible influences on the dispersion curve and displacement and stress mode shapes but evident influences on the attenuation curve and the mode shapes of electric potential, electric displacement, carrier perturbation density and the electric current density.

Semi-analytical IGA-based computation of wave dispersion in fluid-coupled anisotropic poroelastic plates

Understanding dispersion relations and wave mode shapes is vital in nondestructive control of dynamic behaviors of poroelastic composites. In the framework of semi-analytical finite element (SAFE) method, this paper presents two numerical approaches so-called semi-analytical isogeometric Galerkin (SAIGA-G) and semi-analytical isogeometric collocation (SAIGA-C) for computing dispersion of guided-waves in anisotropic poroelastic plates immersed in acoustic fluids. Biot’s theory was used for describing the dynamic behavior of anisotropic poroelastic material. Assuming the structures is homogeneous along its axial direction, the Non-Uniform Rational B-splines (NURBS) was successfully employed in procedures using isogeometric Galerkin and collocation methods. The numeral studies showed that the SAIGA-G method using high continuity NURBS basis allowed to significantly improve the accuracy as well as the convergence rate of the wave dispersion solutions in compared with the conventional SAFE method, which used Lagrange basis functions. Otherwise, the SAIGA-C method was shown to have similar performance in terms of accuracy to the SAFE method.

Analytical and empirical models for the acoustic dispersion relations in partially filled water pipes

Sound propagation in a partially filled pipe is a complex and not a well-researched topic. However, this class of pipes is of high importance because it is typical to millions of kilometres of buried sewers and drainage pipes laid across the world. Good knowledge of conditions in these pipes is critical to their successful operation and proactive rehabilitation. Acoustic methods are attractive means for their rapid condition assessment. Therefore, good understanding of sound propagation in this class of pipes is of importance. In response, this paper studies acoustic waves propagation in a pipe with different water filling ratio. Based on a 2-dimensional finite element method (FEM) study, an analytical approximation and empirical model are developed and proposed to predict the acoustic wave dispersion relations (curves) and mode shapes. These two models can be used to predict the acoustic frequency response function of a partially filled pipe with an accuracy of better than 3 dB below twice the first non-axisymmetric mode frequency, but at 1/2000 of computational cost when compared with the FEM. This work paves the way to the development of highly efficient, real-time robotic systems for autonomous inspection of partially filled pipes with acoustic means.

Waves in a generally anisotropic viscoelastic composite laminated bilayer: Impact of the imperfect interface from perfect to complete delamination

Composite laminates are usually made of anisotropic viscoelastic materials with layer-dependent properties. Although such structures are advantageous over the traditional single-material structure, their interfacial damage/failure is a great concern. In this paper, we present the general formulation and the corresponding solution to enhance our understanding on the guided waves in such laminates with imperfect interfaces. The formulation is based on the modified dual variable and position (DVP) method combined with a powerful and accurate root-finding algorithm. While the former overcomes the numerical instability when handling layered structures, the latter finds all the roots of the dispersion equation in a complex domain providing the full three-dimensional (3D) dispersion curves. After the proposed formulation is validated by the semi-analytical finite-element (SAFE) method for the perfect interface case, the present solution is applied to analyze the dispersion curves and mode shapes for a bilayer laminate with imperfect interface. Numerical examples demonstrate the detailed evolution mechanisms on wave features from perfect interface to complete delamination in the bilayer with imperfect interface. The results presented in this paper provide important insight to guide the in-situ monitoring of the evolution processes of the wave mode shapes and dispersion curves from perfect interface to complete delamination.

Sensitivity analysis of generalised eigenproblems and application to wave and finite element models

The first and second order sensitivity analysis of the eigenvalue problem of generalised, non-symmetric matrices using perturbation theory is developed. These results are then applied to sensitivity analysis of wave propagation in structures modelled using the wave and finite element (WFE) method. Three formulations of the WFE eigenvalue problem are considered: the transfer matrix method, the projection method and Zhong's method. The sensitivities with respect to system parameters of wavenumbers and wave mode shapes are derived. Expressions for the group velocity are presented. Numerical results for a thin beam, a foam core panel and a cross-laminated timber panel are used to demonstrate the proposed approach. It is shown that sensitivities can be calculated at negligible computational cost.

Influence of Surface Conductivity on Dispersion Curves, Mode Shapes, Stress, and Potential for Lamb Waves Propagating in Piezoelectric Plate.

Lamb waves propagating in piezoelectric plate covered by thin-film electrodes with arbitrary conductivity are investigated. The structure consists of three parts, piezoelectric plate, top and bottom thin films with arbitrary conductivities, and top and bottom free space. The governing equations in each part are first solved, and then, continuous conditions are used to form the propagating matrix. Finally, the dispersion equations are obtained by boundary conditions. Calculating the dispersion equations with different conductivities, the dispersion curves in high conductivity and zero conductivity are compared. The wave mode shapes of different branches are investigated, including the distributions of mechanical displacements and stresses in the piezoelectric plate and distributions of electrical displacement and potential in both plate and free space. The relationships between frequency difference and wave mode shapes in zero and high conductivity are investigated. Furthermore, the dispersion curves of general conductivity are calculated. The attenuation of waves is compared in different branches and different conductivities. The wave mode shapes are also calculated and compared with the cases in zero and high conductivities.

Shear horizontal wave propagation along a periodic metal grating surface of a magneto-electro-elastic substrate

This paper analyzes shear horizontal (SH) waves propagating in thin metal strips deposited periodically onto the surface of a magneto-electro-elastic (MEE) substrate. The MEE substrate is assumed to be a transversely isotropic crystal with 6 mm symmetry or a hexagonal (6 mm) crystal. A field analysis method, which is based on the Bloch–Floquet theory and the coupled equations of wave motion, is employed to derive the dispersion equations and wave mode shapes. Numerical examples are presented for an MEE composite (PZT4/CoFe2O4) and an aluminum (Al) strip. For comparison, different constituent ratios of PZT4/CoFe2O4 are considered. The effect of the ratios of strip height and width to grating periodicity on frequencies, phase velocities, as well as wave mode shapes is discussed in detail. The results show that there is a bandgap at the resonance condition when the grating periodicity matches the wavelength of the SH surface wave. This metal grating surface can trap SH surface waves by slowing the SH waves. The wave mode shapes indicate that the SH surface waves are trapped mainly on these metal strips. As the ratio of strip height to grating periodicity increases, the trapping effect is more pronounced and the gap becomes wider. The ratio of strip width to grating periodicity significantly affects the behavior of SH surface waves. The present results are relevant in the application of MEE surface acoustic wave resonators.

A Graphical Analysis Method of Guided Wave Modes in Rails

The cross-section of a rail has a complex geometry, and there are many propagating modes of ultrasonic guided waves in a rail. The analysis of mode shapes or the cross-sectional wave structure is of high significance to the design of an appropriate wave excitation approach for long-range defect detection of a rail. Traditionally, the semi-analytical finite elements (SAFE) method is used to obtain ultrasonic guided waves’ dispersion curves of a rail. Then, through solving the eigenvectors, it is able to calculate the displacement values of discrete nodes in three degrees of freedom (DOFs) and further obtain the wave structures. In this paper, a graphical analysis method of guided wave mode shapes is proposed. The displacements of each node in three DOFs are converted into Red Green Blue (RGB) image pixels, and the complex vibration vector data is expressed by an image. Therefore, the graphical analysis of mode shapes can be realized by using conventional image processing methods without the design of special data processing algorithms. This will improve the processing efficiency, and it is more intuitive and easier to analyze the vibration displacements represented by the image. The simulation results show that the proposed graphical analysis method can quickly and precisely locate the excitation position of the guided wave mode in the rail. By adopting image processing methods, such as the K-means clustering algorithm, the guided wave modes at a 35 kHz frequency in a rail are classified according to their mode shapes. Classification is essential for exploring the relations and fundamentals of vibrations in modes. The graphical analysis method proposed in this paper provides a novel method for the mode analysis of guided waves in rails.

The Computation of Complex Dispersion and Properties of Evanescent Lamb Wave in Functionally Graded Piezoelectric-Piezomagnetic Plates.

Functionally graded piezoelectric-piezomagnetic material (FGPPM), with a gradual variation of the material properties in the desired direction(s), can improve the conversion of energy among mechanical, electric, and magnetic fields. Full dispersion relations and wave mode shapes are vital to understanding dynamic behaviors of structures made of FGPPM. In this paper, an analytic method based on polynomial expansions is proposed to investigate the complex-valued dispersion and the evanescent Lamb wave in FGPPM plates. Comparisons with other related studies are conducted to validate the correctness of the presented method. Characteristics of the guided wave, including propagating modes and evanescent modes, in various FGPPM plates are studied, and three-dimensional full dispersion and attenuation curves are plotted to gain a deeper insight into the nature of the evanescent wave. The influences of the gradient variation on the dispersion and the magneto-electromechanical coupling factor are illustrated. The displacement amplitude and electric potential and magnetic potential distributions are also discussed in detail. The obtained numerical results could be useful to design and optimize different sensors and transducers made of smart piezoelectric and piezomagnetic materials with high performance by adjusting the gradient property.

Accurate characterization of 3D dispersion curves and mode shapes of waves propagating in generally anisotropic viscoelastic/elastic plates

We investigate wave propagation in generally anisotropic viscoelastic plates. The generally anisotropic material system has 21 independent complex coefficients in its stiffness matrix when viscoelasticity is taken into account. The dispersion equation is obtained in the form of determinant vanishing in the complex domain based on the straightforward derivation. An accurate root-searching in the dispersion equation for arbitrary ranges of frequencies and complex wavenumbers, however, is very difficult, if not impossible. In this paper, a novel algorithm is introduced to calculate the 3D dispersion curves. An approximate solution in the low attenuation range via the semi-analytical finite element (SAFE) method is also used to compare and validate the introduced algorithm. Using this algorithm, various peculiar wave features are then investigated in details. These include the attenuation jump and branches exchange in viscoelastic model caused by conversion of wave mode shapes, and the veering of dispersion branches in the corresponding elastic medium. The general anisotropic (viscoelastic/elastic) models are further compared with the isotropic ones to identify the similarity and difference on wave features between them. The proposed accurate algorithm along with the observed features should be particularly useful in nondestructive evaluations via waves in viscoelastic/elastic plates and structures.

A Benchmark Study of Modeling Lamb Wave Scattering by a Through Hole Using a Time-Domain Spectral Element Method

Ultrasonic guided waves (GWs) are being extensively investigated and applied to nondestructive evaluation and structural health monitoring. Guided waves are, under most circumstances, excited in a frequency range up to several hundred kilohertz or megahertz for detecting defect/damage effectively. In this regard, numerical simulation using finite element analysis (FEA) offers a powerful tool to study the interaction between wave and defect/damage. Nevertheless, the simulation, based on linear/quadratic interpolation, may be inaccurate to depict the complex wave mode shape. Moreover, the mass lumping technique used in FEA for diagonalizing mass matrix in the explicit time integration may also undermine the calculation accuracy. In recognition of this, a time domain spectral element method (SEM)—a high-order FEA with Gauss–Lobatto–Legendre (GLL) node distribution and Lobatto quadrature algorithm—is studied to accurately model wave propagation. To start with, a simplified two-dimensional (2D) plane strain model of Lamb wave propagation is developed using SEM. The group velocity of the fundamental antisymmetric mode (A0) is extracted as indicator of accuracy, where SEM exhibits a trend of quick convergence rate and high calculation accuracy (0.03% error). A benchmark study of calculation accuracy and efficiency using SEM is accomplished. To further extend SEM-based simulation to interpret wave propagation in structures of complex geometry, a three-dimensional (3D) SEM model with arbitrary in-plane geometry is developed. Three-dimensional numerical simulation is conducted in which the scattering of A0 mode by a through hole is interrogated, showing a good match with experimental and analytical results.