Rayleigh Method Research Articles

Rayleigh method adapted for the study of the optical response of natural photonic structures

To study the electromagnetic response of natural structures that exhibit interesting optical properties, we developed a computational tool to solve the problem of electromagnetic scattering by a rough interface between two isotropic media, based on the Rayleigh method. The key aspect of the developed formalism is its capability of introducing the interface profile within the code by means of a digitalized image of the structure, which can be either obtained from an electron microscopy image or simply by design according to the complexity of the scattering surface. As application examples, we show the results obtained for surfaces taken directly from microscopy images of two different biological species. This approach constitutes a fundamental step in order to model the electromagnetic response of natural photonic structures.

Acoustic analysis of impact sound on vibrating circular membranes

A finite element method (FEM) - boundary element method (BEM) model is developed to compute the sound generated by of a force acting on a circular membrane (drumhead). A vibro-acoustic analysis that combines modal FEM analysis, a FEM steady state dynamic analysis (SSD), considering harmonic loading and boundary element acoustics, is performed. The drumhead vibrates due to the force impact and the sound is emitted in the air. The vibration of structural response is initially computed, and the obtained results are set to be the boundary conditions of the acoustic analysis in the vibro-acoustic simulation. The radiated sound can be computed at any point of the solution domain. Various materials used by drumhead manufacturers are tested and a parametric analysis focusing on the mesh density of the models is presented. The impact sound and the acoustical characteristics of the simulated test cases are evaluated. The Rayleigh method is also applied to the acoustic simulations and is further compared to the BEM simulation results. The outcomes of this study may be further used as reverse engineering inputs, to machine learning models for the estimation of the physical and mechanical parameters of drumheads from audio signals.

Impact of imperfect corrugated interface in piezoelectric-piezomagnetic composites on reflection and refraction of plane waves.

Defects such as imperfect geometric/mechanical/dielectric/magnetic interfaces affect the performance and life of structural components used in aircraft technology and underwater navigation. The major drawback of an imperfect interface in a system is its disability to transfer stress or potential across the surface effectively. Nondestructive evaluation, particularly P-wave reflection imaging, contributes significantly to detecting defects and in situ conditions of such structural components. The present study encapsulates the reflection/refraction phenomenon of quasi-pressure waves when they strike the mechanically/dieletrically/magnetically imperfect corrugated interfaces of piezoelectric and piezomagnetic half-spaces. The reflected/refracted wave fields not only consist of regular bulk and surface waves but also a spectrum of irregular bulk waves, arising as the result of geometrical irregularities at the interface. The reflection and refraction coefficients (RECs/RACs) of all regular/irregular waves are derived analytically by using Rayleigh's method of slope approximation (first iteration of small perturbation). The existence of the critical angle is interpreted from the slowness curves of the bulk waves. The influences of the corrugation, angular frequency, and interface imperfection on the RECs/RACs are the major findings reported in the study. The reflected/refracted potential waves diminish for the dielectrically/magnetically imperfect interface. The energy coefficients of the reflected/refracted waves are derived, and the energy conservation law is satisfied.

A new modal method for the grating diffraction

A new method whose basis function expressed by that of a interface is proposed to analysis the grating diffraction, which can solve the physical defects of the Rayleigh method and avoid the coordinate transformation of the Coordinate transformation method. This method is named as the Huygens modal method. At the same time, the Huygens modal method can simplify the eigen matrix equation and the traditional Fourier modal method can be summarized as a special case of the Huygens modal method. In the end, a general form of the H-method of a grating diffraction is derived.

Аналитические оценки деформаций и собственных частот опор линий электропередач

The purpose of the study is to derive analytical expressions for estimating the lower vibration frequency of the power line support truss flat models. The forces in the bars of statically determinate structures are determined by the method of cutting out nodes in a program written in the Maple symbolic mathematics language. To find deformations, the Maxwell-Mohr's formula is used under the assumption that all rods are elastic, and that the supports are modeled by rigid rods. It is supposed that the hinges are ideal, and the mass of the structure in the form of point loads is distributed over the truss nodes, and only horizontal load displacements are considered. In comparison with similar problem statements with analytical forms of solution, the present study takes into account the masses at all nodes of the structure. For two-sided estimation of the fundamental frequency, the methods of Dunkerley and Rayleigh are used. The coefficients of the formulas in the solutions obtained for trusses with different numbers of panels form sequences, the common terms of which from the solution of linear recurrent equations give the final formula for the frequency dependence on the number of panels. As a result of the study, formulas for deflection and estimation of the fundamental frequency of truss natural vibration depending on the number of panels and dimensions of the structure have been derived. The obtained formulas can be used in carrying out engineering analyses of power transmission line supports. The formulas for the deflection and frequencies of the studied trusses have a form simple and convenient for use (in particular, for assessing the accuracy of numerical solutions). The frequency obtained by the Rayleigh method is much closer to the fundamental natural frequency than its value estimated using the Dunkerley method.

Mathematical modeling of pressure distribution during deformations of the intervertebral disc

Based on the analysis of the mechanical models of the intervertebral disc given in the literature, it is concluded that the finite element computational grids used in them do not reflect the real structure of the intervertebral disc. In this regard, a mechanical model of the inter- vertebral disc was built, the structure of which is close to its real structure. The proposed model was used to determine the dynamics of the pressure distribution in the intervertebral disc when one of the vertebrae is rotated by a given angle. To determine the resulting bulges of each cell in the structure of the fibrous ring, the Rayleigh method and its modification were used. This made it possible to rationally calculate the volumes of cells. When calculating the pressure in each cell, its linear dependence on the deflection value of the cell was used. As a result of the proposed algorithm, the pressure dynamics in each cell of the intervertebral disc model was determined when the angle between the vertebrae changed.

Analytical Modeling of the Mechanical Behavior of MEMS/NEMS-Multilayered Resonators With Variable Cross-Sections for Sensors and Energy Harvesters

Resonators based on micro and nanoelectromechanical systems (MEMS/NEMS) are used in many applications, including biological and gas sensors, magnetic field sensors, RF switches, accelerometers, piezoelectric micro and nanogenerators, and viscosity sensors. The design of these resonators requires analytical models to predict their mechanical behavior and optimize the sensitivity and resolution. However, most of these models are only applied to resonators with rectangular and uniform cross-sections. In this paper, we present the analytical modeling to determine the first bending resonant frequency, out-of-plane deflections, and normal stresses of MEMS/NEMS-based multilayered resonators with variable cross-sections and multiple fixed supports. The proposed modeling is derived using the well-known Rayleigh and Macaulay methods, as well as the Euler-Bernoulli beam theory. This analytical modeling is applied to four multilayered resonators with different clamped supports and non-uniform cross-sections. The results of our analytical modeling agree well with respect to those of finite element method (FEM) models and experimental data reported in the literature. The proposed analytical modeling can be used to estimate the frequency shift of resonators due to variations of their geometric parameters, number of clamped-supports or mechanical properties of the materials. Furthermore, this modeling can be used to obtain optimal designs of resonators that ensure safe operations and enhanced performance for sensors and energy harvesters in telecommunications, automotive sector, aerospace industry, consumer electronics, non-destructive testing, and navigation.

Applying adjustment factors in Rayleigh method to calculate vibrations principal frequency of a shell with rectangular cross-section

The applying of adjustment factors in Rayleigh method to calculate vibrations principal frequency of a shell with rectangular cross-section is considered in the paper. Behavior patterns of the adjustment factors are generalized. Relationship between the adjustment factors and properties of the approximate formulae is analyzed.

Stiffness Identification Method of Simply Elastic Supported Beam based on First-order Modal Rayleigh Solution

In order to evaluate the support state of bridges with elastic supports, based on first-order mode Rayleigh solution, a stiffness identification method of elastic supports beam is proposed. The superposition displacement of static displacement and rigid body displacement under sinusoidal load is used as the vibration mode function. Based on the mode function, the calculation formula of the fundamental frequency for the simply supported elastic support beam is derived according to the Rayleigh method. The identification method of the support stiffness and beam stiffness of the elastic support beam is derived by using the measured frequency and the mode value. A precast box girder is analyzed by an example. According to the measured results and finite element results, the stiffness of the precast box girder and the stiffness of the laminated rubber bearing are identified by the stiffness identification method established.

Damage Detection of Tension Pendulums in Cable-Stayed Bridges Using Structural Frequency Variance

Structural health monitoring represents an efficient way to diagnose the condition of cable-stayed bridges. Most studies assessing the condition of cable-stayed bridges and detecting damage to them have been developed to monitor the longitudinal performance of bridge bearings. However, few studies have aimed to detect the damage to tension pendulums, which are usually tie-down cables that resist girder uplift from auxiliary piers. This study aims to develop an effective method for detecting the damage of tension pendulums in cable-stayed bridges using structural modal frequencies. First, the sensitivity of the bridge frequency to tension pendulum damage is derived according to the Rayleigh method. Then, a finite-element simulation of a cable-stayed bridge is conducted to validate the rationality of the frequency index. Finally, a case study on tension pendulum damage is carried out for the cable-stayed Yonghe Bridge. The results indicate that the frequency change is far more sensitive to the damage of the tension pendulums than to the damage of the stay cables. The failure of tie-down cables will result in a substantial decrease in the bridge frequency, up to approximately 35% for the first vertical bending mode of the bridge. In practice, the damage of tension pendulums can be effectively detected by a sudden decrease in the measured frequency of cable-stayed bridges.

Electromechanical Modeling of Vibration-Based Piezoelectric Nanogenerator with Multilayered Cross-Section for Low-Power Consumption Devices

Piezoelectric nanogenerators can convert energy from ambient vibrations into electrical energy. In the future, these nanogenerators could substitute conventional electrochemical batteries to supply electrical energy to consumer electronics. The optimal design of nanogenerators is fundamental in order to achieve their best electromechanical behavior. We present the analytical electromechanical modeling of a vibration-based piezoelectric nanogenerator composed of a double-clamped beam with five multilayered cross-sections. This nanogenerator design has a central seismic mass (910 μm thickness) and substrate (125 μm thickness) of polyethylene terephthalate (PET) as well as a zinc oxide film (100 nm thickness) at the bottom of each end. The zinc oxide (ZnO) films have two aluminum electrodes (100 nm thickness) through which the generated electrical energy is extracted. The analytical electromechanical modeling is based on the Rayleigh method, Euler–Bernoulli beam theory and Macaulay method. In addition, finite element method (FEM) models are developed to estimate the electromechanical behavior of the nanogenerator. These FEM models consider air damping at atmospheric pressure and optimum load resistance. The analytical modeling results agree well with respect to those of FEM models. For applications under accelerations in y-direction of 2.50 m/s2 and an optimal load resistance of 32,458 Ω, the maximum output power and output power density of the nanogenerator at resonance (119.9 Hz) are 50.44 μW and 82.36 W/m3, respectively. This nanogenerator could be used to convert the ambient mechanical vibrations into electrical energy and supply low-power consumption devices.

Analytical modeling of mechanical behavior for MEMS/NEMS-based single-clamped multilayer resonators with symmetrical complex shapes

Several micro/nano-electromechanical systems (MEMS/NEMS) devices are composed by complex-shaped multilayer resonators such as energy harvesters, gas and biological sensors, magnetic field sensors, accelerometers, and viscosity sensors. These devices require analytical models to predict the best mechanical performance, improving their sensitivity and resolution. Here, we present the analytical modeling to determine the mechanical behavior of MEMS/NEMS-based single-clamped multilayer resonators with symmetrical complex shapes. This modeling can estimate the first bending resonant frequency and out-plane deflections of multilayer resonators using the Rayleigh and Macaulay methods, as well as the Euler–Bernoulli beam theory. In addition, the quality factors of these multilayer resonators are calculated considering the air damping at atmospheric pressure. Also, finite element method (FEM) models are developed to obtain the mechanical behavior of the resonators. The results of our analytical models agree well with those of FEM models and experimental data reported in the literature. The proposed analytical modeling can be used to enhance the mechanical response of MEMS/NEMS devices formed by multilayer resonators with symmetrical complex cross-sections.

Spatiotemporal characteristics of high-density gas jet and absolute determination of size and density of gas clusters

Properties of gas clusters such as the size and number density when expanding into the vacuum after passing through a conical nozzle are analyzed for argon at an average density of 1020/cm3. Temporally and spatially resolved size and density distribution were measured from all-optical methods of Rayleigh scattering measurement and Nomarski interferometry using a CW laser. At the gas backing pressure of 80 bar, Ar clusters as large as 100 nm were obtained, which differs significantly from the size estimated by the conventional Hagena scaling law. The two independent methods of cluster characterization presented here would be useful to precisely determine the initial conditions in a variety of intense laser-cluster interaction driven applications such as neutron generation, thermonuclear fusion, efficient x-ray emission, and energetic ion acceleration.

An improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term

In this paper, we propose an improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term. The second-order backward differentiation formula (BDF2) and weighted and shifted Grunwald-Letnikov difference (WSGD) formula are employed to discretize first-order time derivative and the time fractional-order derivative, respectively. Moreover, a linearized difference scheme is proposed to approximate the nonlinear source term. Together with the Galerkin finite element method in the space direction, we present a fully discrete scheme for the fractional Rayleigh–Stokes problem with a nonlinear source term. Based on a novel analytical technique, the stability and the convergence accuracy in $$L^{2}$$ -norm with $$O(\tau ^{2}+h^{k+1})$$ are derived in detail, and this convergence order is higher than the previous work. Finally, some numerical examples are presented to validate our theoretical results.

Development and performance of a novel ultrasonic vibration plate sonotrode for grinding

Abstract Compared with Conventional Grinding (CG), the Ultrasonic Vibration-Assisted Grinding (UVAG) is more competitive for the machining of difficult-to-cut materials. In this article, a novel ultrasonic vibration plate sonotrode that enables the special longitudinal full-wave and transverse half-wave vibration modes was proposed. The vibration characteristics of the proposed sonotrode was theoretically studied based on the apparent elasticity and Rayleigh methods, achieving a single longitudinal workpiece ultrasonic vibration with an amplitude of 7.6 μm. A superior vibration uniformity was achieved, as indicated by the coupling coefficient of 0.5. Based on the proposed method, the normal and tangential grinding forces decreased by 35 % and 39 % in comparison with the CG, and an improved machined surface was obtained in the UVAG of Ti-6Al-4V, with the following characteristics: vibration amplification of 7.6 μm; a cut depth of 0.1 mm; a workpiece feed rate of 100 mm/min; and a grinding speed of 30 m/s.

A level set method for simulating wrinkling of extruded viscoelastic sheets

AbstractWhen a polymer is extruded freely from a rectangular die of large cross‐sectional aspect ratio, wrinkles are observed. While not present in extruded Newtonian materials, such wrinkles develop in extruded viscoelastic sheets and are understood as an elastic stress‐driven instability. The present study is devoted in developing a transient finite element method, which combines the matrix‐logarithm‐based formulation of the conformation tensor and the single‐phase level set method, for simulating wrinkles that form during sheet extrusion of viscoelastic fluids. Numerical analyses of sheet extrusion were conducted over a wide range of flow rate and width‐to‐thickness ratio of the die exit cross section, χ, to determine critical conditions for the onset of wrinkling of extruded sheets. For large aspect ratios, that is, χ >> 1, wrinkles develop at moderate extrusion flow rate, corresponding to a Weissenberg number of about 29. Calculations based on Rayleigh's energy method show that the critical compressive stress, σc, for the onset of wrinkling of an elastic sheet scales like σc~1/χ2, with a significant drop for χ >> 1. As next to the die exit lip, compressive normal stresses are induced in the extruded sheet, wrinkling will take place for large χ (σc being small), in accordance with numerical predictions.

Study of bandgap property of a bilayer membrane-type metamaterial applied on a thin plate

This work is aimed to study the bandgap property of a thin plate structure with periodically attached bilayer membrane-type resonators. An analytical method based on the Plane Wave Expansion (PWE) method combined with the Rayleigh method, is proposed to predict the bandgap property of bilayer membrane-type metamaterials. The accuracy of the proposed method is verified by the finite element analysis, and a parametric analysis is conducted to reveal the effect of parameters on the bandgap performance. It is found that such a metamaterial can generate two separated bandgaps through the contribution of its two layers of membranes. It is observed that the increase of membrane tensile stress or the magnitude of attached mass can lead to the broadening of bandgaps, whilst the change of unit cell's periodicity has the opposite effect. In addition, if compared with the corresponding single layer membrane-type metamaterials, it is shown that the bilayer membrane-type's first bandgap is suppressed while the second one is extended. However, by applying proper membrane tensile stress and mass magnitude, the suppression of the first bandgap can be weakened whilst allowing the tuning of the bandgap location. These characteristics reveal the benefits of using bilayer membrane-type metamaterial as it possesses higher agility in bandgap tuning. The proposed method can provide an effective tool for the bilayer membrane-type metamaterial design and optimisation.

Observation of Selényi rings in the diffraction patterns of layered samples with periodic arrays of cylindrical structures.

We present an experimental and numerical study of the optical properties of nanofabricated samples with layered dielectric structures. The samples, which contain periodic arrays of silicon disks over a flat layer of silicon dioxide on a silicon substrate, present diffraction and thin film interference effects. Well-defined circular fringes that modulate the intensity of the diffraction orders are observed in the far-field angular distribution of scattered light. We also find that although the angle of incidence modulates the intensity of the observed circular ring patterns, it has little or no effect on their angular position. The problem is modeled theoretically through numerical calculations based on a Rayleigh method.

Optical, stability and energy performance of water-based MXene nanofluids in hybrid PV/thermal solar systems

Solar thermal collectors have been recognized as promising devices for solar energy harvesting. The absorbing properties of the working fluid are crucial because they can significantly influence the efficiency of the solar thermal collectors. The performance of photovoltaic-thermal (PV/T) systems can be optimized by applying nanofluids as working fluids. MXene is a newly developed 2-D nanomaterial that has proven excellent potential in electrical applications with a lack of research in the thermal and optical applications. The present work extensively studied the optical potential of the water/MXene nanofluids with respect to the variation of MXene concentrations (0.0005–0.05 wt%) and types of surfactant (CTAB or SDBS) used in a hybrid PV/T system. The relationship between the investigated parameters was evaluated through an experimental correlation. The evaluation of the nanofluids in term of the transmittance was conducted through the Rayleigh method. The MXene concentrations and the types of the surfactant play predominant role in the transmittance, absorbance and dispersion stability of the water/MXene nanofluids. The corresponding effects due to these factors become the most noticeable in the wavelengths of 300–1350 nm. Low concentration of the MXene and shorter path lengths lead to higher transmittance. The application of the low concentration of water/MXene nanofluids as the optical filtration in a hybrid PV/T system yields a higher performance compared to a conventional PV/T system. Therefore, this research work provides novelty value in understanding the impacts of using water/MXene nanofluid in the hybrid PV/T solar collectors to harness additional energy.

Application of the Ritz–Rayleigh method for Lamb waves in extremely anisotropic media

We propose a numerical approach for the calculation of frequency–dispersion curves in a flat anisotropic waveguide based on the Ritz–Rayleigh method, offering several significant benefits over commonly used analytical and numerical models. The problem is linearized using a tailored functional basis based on Legendre polynomials, utilizing all symmetries provided by the problem to reduce the computational demands, which brings significant benefits for an inverse procedure, and thus for material characterization. The approach is completely general in terms of elastic properties and direction of propagation. As an exemplary calculation, frequency–dispersion curves of phase and group velocity are calculated for a (001)-oriented free-standing film of the Ni–Mn–Ga alloy exhibiting a strong cubic anisotropy. In the case of propagation along the [100] direction, the dispersion curves oscillate strongly, creating pairs of symmetric and antisymmetric modes. Taking advantage of the generality of the approach, the curves are also presented for the case of extreme anisotropy. Sets of curves for directions [110] and [1 0.9 0] (i.e., 3∘ off [110]), where the pseudo-surface wave exists, are also calculated. High-frequency asymptotes are shown to correspond with surface and bulk modes propagating along the plate.