Free Wave Propagation Research Articles

Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity

In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship’s deck, an aeroplane fuselage, a satellite antenna, a wall panel. To avoid transmission of the vibration and structure-borne sound, radial or polar periodicity may be used. In these cases, numerical approaches to study free and forced wave propagation close to the excitation source in polar coordinates are desirable. This is the paper’s aim, where a numerical method based on Floquet-theory and the FE discretision of a finite slice of the radial periodic structure is presented and verified. Only a small slice of the structure is analysed, which is approximated using piecewise Cartesian segments. Wave characteristics in each segment are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis, while wave amplitude change due to the changes in the geometry of the slice is accommodated in the model assuming that the energy flow through the segments is the same. Forced response of the structure is then evaluated in the wave domain. Results are verified for an infinite isotropic thin plate excited by a point harmonic force. A plate with a periodic radial change of thickness is then studied. Free waves propagation are shown, and the forced response in the nearfield is evaluated, showing the validity of the method and the computational advantage compared to FE harmonic analysis for infinite structures.

Influence of attached inertia and resonator on the free wave propagation in 2D square frame grid lattice metamaterial

This paper presents spectral element based approach for determining free planner wave propagation in two-dimensional periodic square-frame-grid-lattices (SqL). Implementing the newly developed nonlinear eigenvalue solver in conjunction with Bloch–Floquet periodic boundary condition, band structures for various configurations of SqL with attached mass or spring-mass resonator are analysed. Variation of the attenuation bandwidth is studied for different mass and frequencies of the attachments to the SqL. At the natural frequency of the resonator, an attenuation bandgap can be perceived; therefore, by tuning the natural frequency of the resonator, sub-wavelength band-gap can be obtained at the desired frequency for enhanced vibration or noise isolation. However, in few cases, mostly while the resonators are attached at the center point of the SqL, the attenuation bandgap near the natural frequency of the resonator may disappear as the center point lying in the node for that free wave frequency. Additionally, existence of dirac cone, eigenfrequency-loci veering are observed in the band-structures. The iso-frequency contours are also investigated to develop an insight comprehension about the underlying physics of energy transmission and mechanics of wave propagation. Various salient wave propagation features, namely self-collimation, lensing, and negative refraction of group velocity are identified and elucidated in this paper.

Investigation of the propagation of free waves in a rod in the design of building structures

This article investigates the dispersion dependences of waves propagating in a system consisting of an elastic rod immersed in a linear elastic medium. The phase velocities of a wave with a length λ, which propagate under stationary conditions, are investigated. It was found that for a sufficiently small η (a/λ), i.e., for relatively large wavelengths, no waves can propagate in the system under stationary conditions, i.e. there are critical values of η below which no real values of the phase velocity are possible.

Wave-Mechanical Electron-Optical Modeling of Field-Emission Electron Sources

Electron source coherence has a very important influence on the imaging capabilities of modern electron microscopes. However, conventional electron source models that are based on geometrical electron optics implicitly assume that the emission from the source surface is fully incoherent, which can complicate the treatment of highly coherent field-emission sources. In an attempt to treat the wave-optical properties of electron sources, models inspired by light optics treatments of (partially) coherent sources, which assume a planar source and free wave propagation, have been developed. In this case the underlying assumptions are problematic, because the source surface of a field emitter can have a radius of curvature on the nanometer scale, and the emitted electrons are accelerated by a strong, inhomogeneous electrostatic field following emission. We introduce a model based on wave-mechanical electron optics that draws on a quantum mechanical description of electron emission and propagation to obtain a physically consistent treatment of the wave-mechanical properties of electron sources. We apply the model to investigate spatial resolution limits in low-energy electron holography and microscopy, where it is shown that aberrations and coherence properties of the electron source are crucial and interrelated. The wave-mechanical electron-optical model can, furthermore, be readily generalized to assess and improve electron source performance in other scenarios and techniques where spatial and temporal coherence, and electron-optical aberrations, are relevant.

Sound radiation analysis with an enriched Timoshenko beam model based on second strain gradient theory

In this work, an enriched Timoshenko beam model describing non-classical bending and shear motion is employed to study its free wave propagation, frequency response, and sound radiation while considering the local behavior of its internal heterogeneity. The enriched Timoshenko beam model is established based on Mindlin’s Second Strain Gradient (SSG) theory, which can capture the long-range interactions between underlying micro-structures in the frame of continuum mechanics. Dispersive behaviors of non-classical bending waves, shear waves, and the extra evanescent waves appearing exclusively in the SSG theory model are analyzed. Then investigations are carried forward into the modal density of non-classical bending waves, shear waves, and energy flow through the enriched model. At last, the frequency response and sound radiation of the Timoshenko beam are studied using wave approach based on its multi-modes character. The resulting non-classical frequency response and radiation features are compared with the classical elasticity theory result, upon which, impacts of local behavior in complex media on the structural dynamics and radiation features are discussed. The role of local behavior is, among others, clearly shown in the frame of the non-classical structural behavior.

Propagation and attenuation characteristics of free flexural waves in multi-stepped periodic beams by the method of reverberation-ray matrix

This paper aims to present better physical understanding of flexural wavenumber spectral behaviors of multi-stepped periodic beams by the method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory. According to the Bloch theorem, the formulation of MRRM is presented and the complex flexural wavenumbers of multi-stepped periodic beams are solved by numerical techniques. The validation of MRRM in analyzing flexural wave behaviors of multi-stepped periodic beams is verified by comparison with the analytical wavenumber spectrum of a homogeneous beam. Numerical examples are provided to illustrate the general wavenumber spectral characteristics of an infinite periodic beam with alternate variation of cross-sections. The effects of the material properties and geometrical parameters of the beam segments are evaluated. The most remarkable finding inspired from the wavenumber spectral curves is that the real wavenumber of the multi-stepped periodic beam does not vary with frequency in the attenuation bands, which indicates that the wavelength of the flexural wave is constant in the attenuation bands. Highlights A multi-stepped periodic beam model with arbitrary finite number of beam segments is established. Complex wavenumber spectrum analysis is performed to reveal the propagation and attenuation behaviors of multi-stepped periodic beams. Similarity and comparative studies on the propagation and attenuation characteristics of beams of a uniform cross-section and of variable cross-sections are carried out to provide better physical understanding. An innovative finding that the wavelength of the flexural wave is constant in the attenuation bands is achieved.

Free vibration characteristics and wave propagation analysis in nonlocal functionally graded cylindrical nanoshell using wave-based method

Free vibration characteristics and wave propagation analysis in nonlocal functionally graded cylindrical nanoshell using wave-based method

Calculating the response of waveguides to base excitation using the wave and finite element method

The prediction of the response of waveguides to time-harmonic base excitations has many applications in mechanical, aerospace and civil engineering. The response to base excitations can be obtained analytically for simple waveguides only. For general waveguides, the response to time-harmonic base excitations can be obtained using the finite element method. In this study, we present a wave and finite element approach to calculate the response of waveguides to time-harmonic base excitations. The wave and finite element method is used to model free wave propagation in the waveguide, and these characteristics are then used to find the amplitude of excited waves in the waveguide. Reflection matrices at the boundaries of the waveguide are then used to find the amplitude of the travelling waves in the waveguide and subsequently the response of the waveguide. This includes the displacement and stress frequency response transfer functions. Numerical examples are presented to demonstrate the approach and to discuss the numerical efficiency of the proposed method.

Influence of conicity on the free wave propagation in symmetric tapered periodic beam

In this study, the attenuation characteristics of the unit cell of an infinitely long double symmetric tapered beam (DSTB) is examined. The exact flexural shape functions of the DSTB is derived using the Bessel functions. The band-structures for rectangular and circular solid section with varying conicity is evaluated implementing the transfer matrix method. It is found that the tapered beam produces wider band gaps for an ideal selection of conicity for both rectangular and circular cross-section. It is also evidenced in this paper that the circular cross-section produces wider band gaps with higher level of attenuation in comparison to the rectangular section. Specifically, the first normalized attenuation bandwidth of 165% can be achieved in case of circular section and 97% in case of rectangular section. A representative cantilever finite beam structure comprised of the periodic DSTB also elucidates the reduction in flexural vibration for the same frequency ranges obtained from the dispersion curve of the infinite structure’s unit cell.

A wave and finite element method for calculating sound transmission through rectangular panels

A wave based methodology is proposed to investigate sound transmission through rectangular, rigidly-baffled panels, including the modal behaviour of the structure. The panel is homogeneous in-plane, but the through-thickness construction can be of arbitrary complexity. The fluids are modelled analytically, the structure by the wave and finite element method: a small segment of the structure is meshed to predict the behaviour of an infinite panel with the same properties. The transverse displacement of the panel is modelled as a sum of sine terms, each associated with a specific mode. Neglecting fluid loading, the blocked pressure of the incident wave is decomposed into space harmonics with given trace wavenumbers. An equivalent, infinite plate, excited by a periodic continuation of the loading on the finite panel, is considered. The natural frequencies of the finite panel follow from free wave propagation, its response estimated from that of the infinite panel. The far-field acoustic pressure and sound transmission loss are presented. Numerical results agree well with analytical solutions, other numerical solutions and measurements. The proposed method takes the modal effects fully into account, and is efficient and straightforward to implement.

Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators

Beam lattice materials are characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. The dispersion properties governing the free dynamic propagation of elastic waves can be studied by formulating parametric discrete models of the cellular microstructure and applying the Floquet-Bloch theory. Within this framework, governing the wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical and applied interest. Specifically, the wave propagation can be inhibited by purposely designing the microstructural parameters to open stop bands in the material spectrum at target center frequencies. Based on these motivations, a general dynamic formulation is presented for determining the dispersion properties of mechanical metamaterials, modeled as locally resonant beam lattices with generic coordination number. The mechanism of local resonance is realized by tuning periodic auxiliary oscillators, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent viscoelastic coefficients are obtained by applying the in-space Z-transform and in-time bilateral Laplace transform. The complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed for the beam lattice characterized by quadrilateral periodic cell. The spectral branches may exceed the model dimension, due to the occurrence of pure-damping spectral components. Particularly, the spectra corresponding to Laurent series approximations of the viscoelastic coefficients are investigated and the solution admissibility and convergence for increasing order series is analyzed. The standard dynamic equations with linear viscous damping are recovered at the first-order approximation. Low-order approximations are found to underestimate the real and imaginary parts of the spectrum, as well as the low-frequency stop bandwidth. Finally, the forced response to a harmonic mono-frequent external point excitation is investigated. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint.

Design of tunable acoustic metamaterials with periodic piezoelectric microstructure

An innovative special class of tunable periodic metamaterials is designed, suitable for realizing high-performance acoustic filters. The metamaterial is made up of a phononic crystal coupled to local resonators. Such local resonators consist of masses enclosed into piezoelectric rings, shunted by either dissipative or non-dissipative electrical circuit. By tuning the impedance/admittance of such electrical circuits, it is possible to fully adjust the constitutive properties of the shunting piezoelectric material. This feature paves the way for unconventional behaviours, well beyond the capabilities achievable with classical materials. It follows that the acoustic properties of the periodic metamaterial can be adaptively modified, in turn, opening new possibilities for the control of pass and stop bands. By exploiting a generalization of the Floquet–Bloch theory, the in-plane free wave propagation in the tunable metamaterial is investigated, by varying a certain tuning parameter, to show the efficiency of the proposed shunting piezoelectric system as a wave propagation control device. Particular attention is devoted to the determination of the in-plane constitutive equations of the shunting piezoelectric phase in the transformed Laplace space. Finally, broad design directions of tunable acoustic filters aiming to a changing performance requirement in real-time, is also provided.

High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves

In this paper, we propose a unified and high order accurate fully-discrete one-step ADER Discontinuous Galerkin method for the simulation of linear seismic waves in the sea bottom that are generated by the propagation of free surface water waves. In particular, a hyperbolic reformulation of the Serre-Green-Naghdi model for nonlinear dispersive free surface flows is coupled with a first order velocity-stress formulation for linear elastic wave propagation in the sea bottom. To this end, Cartesian non-conforming meshes are defined in the solid and fluid domains and the coupling is achieved by an appropriate time-dependent pressure boundary condition in the three-dimensional domain for the elastic wave propagation, where the pressure is a combination of hydrostatic and non-hydrostatic pressure in the water column above the sea bottom. The use of a first order hyperbolic reformulation of the nonlinear dispersive free surface flow model leads to a straightforward coupling with the linear seismic wave equations, which are also written in first order hyperbolic form. It furthermore allows the use of explicit time integrators with a rather generous CFL-type time step restriction associated with the dispersive water waves, compared to numerical schemes applied to classical dispersive models that contain higher order derivatives and typically require implicit solvers. Since the two systems that describe the seismic waves and the free surface water waves are written in the same form of a first order hyperbolic system they can also be efficiently solved in a unique numerical framework. In this paper we choose the family of arbitrary high order accurate discontinuous Galerkin finite element schemes, which have already shown to be suitable for the numerical simulation of wave propagation problems. The developed methodology is carefully assessed by first considering several benchmarks for each system separately, i.e. in the framework of linear elasticity and non-hydrostatic free surface flows, showing a good agreement with exact and numerical reference solutions. Finally, also coupled test cases are addressed. Throughout this paper we assume the elastic deformations in the solid to be sufficiently small so that their influence on the free surface water waves can be neglected.

A Comparison of Different Wave Modelling Techniques in An Open-Source Hydrodynamic Framework

Modern design for marine and coastal activities places increasing focus on numerical simulations. Several numerical wave models have been developed in the past few decades with various techniques and assumptions. Those numerical models have their own advantages and disadvantages. The proper choice of the most useful numerical tool depends on the understanding of the validity and limitations of each model. In the past years, REEF3D has been developed into an open-source hydrodynamic numerical toolbox that consists of several modules based on the Navier–Stokes equations, the shallow water equations and the fully nonlinear potential theory. All modules share a common numerical basis which consists of rectilinear grids with an immersed boundary method, high-order finite differences and high-performance computing capabilities. The numerical wave tank of REEF3D utilises a relaxation method to generate waves at the inlet and dissipate them at the numerical beach. In combination with the choice of the numerical grid and discretisation methods, high accuracy and stability can be achieved for the calculation of free surface wave propagation and transformation. The comparison among those models provide an objective overview of the different wave modelling techniques in terms of their numerical performance as well as validity. The performance of the different modules is validated and compared using several benchmark cases. They range from simple propagations of regular waves to three-dimensional wave breaking over a changing bathymetry. The diversity of the test cases help with an educated choice of wave models for different scenarios.

Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams

The transverse free vibration and wave propagation of functionally graded microbeams with an axial motion are investigated based on a nonlocal theory and the Timoshenko beam model. It is assumed the material properties of functionally graded microbeams vary along the thickness direction. The neutral plane of functionally graded materials is introduced, and the inhomogeneity of functionally graded Timoshenko microbeams is considered. The governing equations are derived using Hamilton’s principle, and the differential quadrature method is utilized to determine the first three-order natural frequencies of the microbeams with simply supported and clamped boundary conditions, respectively. The effects of gradient index, nonlocal parameter and axial velocity on natural frequencies are investigated. Moreover, the wave propagation characteristics of functionally graded Timoshenko microbeams are analyzed, and the significant influences of wave number and other variables on wave propagation frequencies and wave velocities are studied. The different influence patterns of nonlocal effect are observed in transverse vibration and wave propagation. The nonlocality shows a weakening phenomenon in transverse vibration of axially moving functionally graded Timoshenko microbeams, while it reveals both weakening and strengthening phenomena in wave propagation. Therefore, two kinds of existing nonlocal scale effects are further confirmed and this is an additional contribution of the present paper.

Wave propagation and scattering in reinforced concrete beams

Steel reinforcement bars (rebars) are vital to the strength of reinforced concrete (RC) structures, but can become damaged due to corrosion. Such damage is generally invisible and non-destructive testing methods are needed to assess their integrity. Guided wave methods are popular because they are capable of detecting damage using sensors placed remotely from the damage site, which is often unknown. This paper predicts free wave propagation in RC beams from which the concept of a guided wave based damage detection method emerges. The wave solutions are obtained using the wave finite element framework where a short section of a beam's cross section is modeled in conventional finite element (FE) and periodic boundary conditions are subsequently applied. Reinforcement elements are used in the FE model of the cross section as a neat and efficient means of coupling the concrete to the rebars and imposing prestress. The results show that prestress, important for static behavior, has a negligible effect on wave dispersion. A RC beam with a damaged section is modeled by coupling three waveguides, the center waveguide being identical to the outer ones except for a thickness loss in one rebar. Only small differences in cut-on frequencies are observed between the damaged and undamaged sections. However, these small differences give rise to strong reflection of some waves at frequencies close to cut-on. Below cut-on, most incident power is transmitted but experiences wave mode conversion, whereas above cut-on most power is transmitted to the same wave type. These observations form the basis for ongoing work to develop a damage detection technique premised on wave reflection near cut-on.

Generalized Transfer Matrix Method for periodic planar media

Sound transmission through infinite planar multilayered structures characterized by in-plane periodicity is accurately and efficiently predicted by exploiting free wave propagation and Bloch modes. A through-thickness transfer matrix is derived for each layer by manipulating the dynamic stiffness matrix related to a finite element model of a unit cell. The transfer matrices of all the layers composing the structure account for the Bloch modes generated in heterogeneous layers. The proposed technique is equally appealing for in-plane homogeneous structures since few elements and no Bloch modes are needed in this case, ensuring high efficiency. In such a framework, the acoustic radiation or transmission of multilayered systems excited by an oblique plane wave can be assessed. The proposed approach is validated in case of structures consisting of heterogeneous layers by comparison with alternative approaches.

Seismic Response and Vibration Transmission Characteristics of Laminated Rubber Bearings with Single Disorder

AbstractThe characteristic receptance method of analyzing the free wave propagation of a finite monocoupled periodic system with a single disorder is applied to study the vibration transmission thr...

Numerical investigation of the potential of tailored inclusions as noise reduction measures

The reduction of noise and vibrations is an important task in automotive and aircraft industry. Various applications require slender or adjustable reduction measures. Metamaterials formed by a periodic assembly of unit cells consisting of non-homogeneous material decomposition and having topology dependent favorable absorption properties can meet this need. The absorption efficiency of different types of metamaterials has already been proven by various authors. Thereby metamaterials formed by a periodic assembly of tuned mass dampers on plate-like structures or porous layers with rigid inclusions have been investigated. Beyond that, we suggest to design metamaterials consisting of inclusions containing appropriately tuned mass dampers embedded in a material layer and to examine their noise reduction potential. It is investigated, whether besides absorbing energy by exciting the damping device, the orientation of the tuned mass damper is active by purposefully deflecting waves inside the structure. The analysis is carried out applying the Wave Finite Element Method. Using the inverse approach of the Wave Finite Element Method, stop bands, i.e. frequencies for which no free wave propagation is observed, are detected. In a parameter study, the influence of the characteristics of the inclusion on the absorption behavior of the metamaterial is determined.

Numerical computation of the spatial decaying wave characteristics for the design of locally resonant acoustic metamaterials

In structural dynamics, periodic resonant inclusions are widely used to improve the vibro-acoustic properties of lightweight structures. Depending on the design of these inclusions, the wave propagation in the structure is modified for specific frequency ranges and stop bands, where in theory no free wave propagation is possible, can be generated. The Wave Finite Element Method (WFEM) investigates the wave propagation in periodic structures based on a Finite Element (FE) model of a single unit cell. Using the direct approach of the WFEM, it is possible to depict the wave characteristics in the frequency range of the resonant frequency of the inclusions. Consequently, a spatial decay for the different wave types can be determined in the frequency range of the potential stop band. The frequency dependent decay characteristics can be used to evaluate the performance of the resonant inclusions and find optimal parameters for the desired application of the metamaterial. The procedure is applied to design beam-like resonators and find an optimal resonator spacing for a beam-like acoustic metamaterial. For a fixed percentage of added mass, the spacing of the resonators can be adjusted to optimize the maximum spatial decay or the stop band size.