Classical Transfer Matrix Method Research Articles

Model refinements with experimental validation in piezoelectric plate actuators

The axisymmetric vibration analysis of circular plates with piezoelectric layers and step changes in plate thickness is conducted. For the chosen configurations, it is necessary to solve a coupled system of differential equations for extensional and flexural vibration to obtain accurate modal results. The adopted computational approach is based upon the classical transfer matrix method. Instead of employing analytical solutions in the form of e.g. Bessel functions, matrix exponentials of the system matrix are directly computed in the transfer matrices for finding highly accurate solutions. The developed approach has enabled the complete quantitative modal analysis in piezoelectric actuators. The procedure is firstly verified against available analytical results for natural frequencies and modes shapes in circular homogeneous plates, and by using finite element simulations for plates with asymmetric step changes in thickness. Ultimately, the developed approach is validated against experimental results of relevance to applications in piezoelectric synthetic jet actuation. The obtained numerical results for modal parameters and frequency response functions are in excellent agreement with the available analytical results, and with previous and newly presented in the paper experimental data, when considering the inherent uncertainty in parameter values. Based on extensive experimental results, realistic damping is introduced in the model of piezoelectric plate actuators. Importantly, it is shown that solutions of high accuracy can be obtained by a more general coordinate system than that traditionally linked to the plate neutral plane(s).

Planar piezoelectric metamaterials: Sound transmission and applicable frequency range in oblique incidence

Frequency limit and applicable frequency range of planar piezoelectric metamaterials connected to external circuits have not been well defined in estimating the sound transmission loss. This article extends the classical transfer matrix method for use in evaluating the sound transmission of thin-plate piezoelectric metamaterials in oblique incidence. Using the Kirchhoff thin plate theory, a modified transfer matrix method that takes factors of external circuits into consideration is developed for attenuation and control of acoustic waves. Several vibro-acoustic analytical models are compared, including the Kirchhoff thin plate theory, the Reissner–Mindlin thick plate theory and the theory of wave propagation in elastic solids. These theories are used to determine the dispersion relation, coincidence and transition frequency of thin and thick plate theories in analysing piezoelectric acoustic metamaterials, alongside a validation using the finite element method. Then acoustic properties of piezoelectric plates connected to passive external circuits are studied parametrically with both dimensional and dimensionless variables based on an equivalent Kirchhoff plate approximation. The findings show that external electrical impedance alone can be used to adjust the resonance frequency over a broad range and thereby control sound transmission loss. This provides considerable flexibility in modifying the acoustic properties of the piezoelectric metamaterial in comparison to traditional, fixed-structure metamaterials. The study indicates a straightforward and powerful analytical approach for the optimization of acoustic insulation using thin-plate piezoelectric metamaterials.

Homogenization and analysis of planar piezoelectric metamaterials: A comprehensive study

This article presents a comprehensive investigation into the homogenization and analysis of planar piezoelectric metamaterials. The classical transfer matrix method is extended by using homogenization methods to evaluate the sound transmission of piezoelectric metamaterials in both normal and oblique incidences. This has been validated numerically using the finite element method and experimentally in the solid medium propagation in normal incidence. Several vibro-acoustic analytical models are compared in oblique incidence, including the Kirchhoff thin plate theory, the Reissner-Mindlin thick plate theory, and the theory of wave propagation in elastic solids. These theories are used to determine the dispersion relation, coincidence, and transition frequency of thin and thick plate theories in analyzing piezoelectric metamaterials. Additionally, acoustic properties of piezoelectric layers connected to external circuits are parametrically studied, exploring both dimensional and dimensionless variables. Results indicate that significant control over the resonance frequency and sound transmission can be enabled by adjusting the external electrical impedance, flat-layer structures, and piezoelectric materials. This demonstrates excellent tunability and compactness of planar piezoelectric metamaterials for space-sensitive applications. The study indicates a straightforward and powerful analytical approach for the optimization of acoustic insulation using planar piezoelectric metamaterials.

A novel generalized sandwich-type replacement beam for static analysis of tall buildings: Inclusion of local shear deformation of walls

As the length of the shear walls increases, local shear deformation leads to increased error when using classical continuum models for static structural analysis of tall buildings. In this paper, a novel continuous beam of the generalized sandwich type is proposed for the structural analysis of tall buildings that have shear walls of significant length as lateral bracing systems. The proposed continuous beam includes in its formulation an additional kinematic field of rotation due to the local shear deformation of the walls and results from the series coupling of a classic sandwich beam and a shear beam. The equilibrium equations, the constitutive laws, and the boundary conditions of the continuous model are obtained with a variational approach using the Hamilton principle on the relevant Lagrange function. A system of three coupled differential equations leading to sixth-order differential equation and six boundary conditions are obtained. Two closed-form analytical methods are proposed for uniform-height tall buildings. For the case of buildings with variable properties along their height, a numerical method of the modified transfer matrix type is proposed, which is directly derived from the analytical method and avoids the need to calculate the inverse of the classical zero transfer matrix method. The numerical results show the validation of the continuous model and the proposed solution methods. In addition, a parametric analysis of 352 buildings was performed to demonstrate the superiority of the continuous model and the proposed solution methods.

Riccati transfer equations for fluid structure interaction in liquid-filled piping systems

In this paper, based on the Riccati transfer matrix method (RTMM), the Riccati fluid structure interaction transfer equations (FSIRTE) are established to improve the numerical stability of the classical fluid structure interaction transfer matrix method (FSITMM). Combined with numerical algorithms for eliminating the singularity points of the Riccati equations, the spare root problem in the calculation process is solved. This method can be used for the natural frequency calculation of liquid-filled piping systems. Compared with finite element method (FEM), it has the characteristics of high calculation efficiency; meanwhile, good numerical stability, compared with FSITMM; and accurate calculation results, compared with method of characteristics (MOC). Numerical simulation results of typical classical examples are given.

An efficient technique in transfer matrix method for beam-like structures vibration analysis

Rotating beam-like structures are widely used in various engineering fields, and the transfer matrix method is a classic analytical method for vibration analysis. Due to eliminating numerical difficulties, Riccati transfer matrix method has been applied. Since this method demands an inverse matrix procedure, a novel algorithm is presented to reduce the matrix order with less computation time. The proposed technique is performed for the vibration analysis of the rotary beam-like structures to characterize the frequencies without limitations from the rotational speed. Rather than numerical problems in the classical approach, this new technique exhibits an improved performance while the key factor for efficiency is the computation time. Comparing the results achieved through the finite element and Rayleigh–Ritz methods shows an acceptable agreement with high accuracy responses. In addition, the computation time for Riccati transformation adoption and the classical transfer matrix method is investigated. This evaluation demonstrates the priority of using this technique in terms of computation cost.

Higher order interface conditions for piezoelectric spherical hollow composites: asymptotic approach and transfer matrix homogenization method

The paper describes the mechanical behavior of composites made of piezoelectric spheres in perfect or imperfect contact. The imperfect contact is achieved by interposing piezoelectric thin adhesive layers between the spheres. First, using asymptotic analysis, transmission conditions of imperfect interface equivalent to the behavior of piezoelectric adhesive layers are obtained at order 0 and 1. These transmission conditions are calculated for ”hard” adhesives, i.e. adhesive materials whose electromechanical constants do not rescale with their thickness. Next, under the assumption of spherical symmetry, the transmission conditions are condensed to a general law of imperfect contact, able to simultaneously describe different contact regimes: piezoelectric hard (order 0 and 1) and soft (or spring-type, order 0 and 1) interface conditions, the perfect continuity conditions, and the piezoelectric rigid (Gurtin–Murdoch or membrane-type) conditions. Lastly, following Bufler’s approach, the homogenization problem of a spherical hollow piezoelectric assembly is solved, extending the classical transfer matrix method to take into account the presence of thin adhesive layers described using the proposed transmission conditions of imperfect contact. A simple numerical example is provided, illustrating the correctness and effectiveness of the homogenization approach in describing the electromechanical behavior of spherical piezoelectric assemblies.

Thermo-optic response of MEH-PPV films incorporated to monolithic Fabry-Perot microresonators

Poly[2-methoxy-5-(2′-ethylhexyloxy)-1,4-phenylene vinylene] (MEH-PPV) is a semiconducting optically active polymer widely used in optoelectronics research. MEH-PPV can be commercially acquired in a large range of molecular weights. However, the influence of this property on the optical performance of the polymer is often disregarded. In this paper, the thermal dependence of the refractive index of MEH-PPV thin films prepared from high and medium molecular weight polymers is investigated. Thus, monolithic Fabry-Perot (FP) microcavities are fabricated, in which the active polymer film is part of their defect layer. It is found that when these devices are used as optical temperature sensors, the position of the emission band of the microcavities excited with a blue diode laser shifts to lower wavelengths when temperature increases with sensitivities in the 0.2–0.3 nm/°C range. This effect is ascribed to the variation in the refractive index of the polymer active layer within the resonator with temperature. According to theoretical simulations of optical transmittance by classical transfer-matrix method and the evaluation of the optical eigenmodes by finite element methods of the manufactured FP resonator cavities, it is found that the MEH-PPV films present negative thermo-optic coefficients of about −0.018 K−1 and −0.0022 K−1 for high and medium molecular weight polymers, respectively, in the temperature range between 20 and 60 °C. These values are about the highest reported so far, to the best of our knowledge, and points to high performance thermal sensor applications.

Molecular polaritons for controlling chemistry with quantum optics.

This is a tutorial-style introduction to the field of molecular polaritons. We describe the basic physical principles and consequences of strong light-matter coupling common to molecular ensembles embedded in UV-visible or infrared cavities. Using a microscopic quantum electrodynamics formulation, we discuss the competition between the collective cooperative dipolar response of a molecular ensemble and local dynamical processes that molecules typically undergo, including chemical reactions. We highlight some of the observable consequences of this competition between local and collective effects in linear transmission spectroscopy, including the formal equivalence between quantum mechanical theory and the classical transfer matrix method, under specific conditions of molecular density and indistinguishability. We also overview recent experimental and theoretical developments on strong and ultrastrong coupling with electronic and vibrational transitions, with a special focus on cavity-modified chemistry and infrared spectroscopy under vibrational strong coupling. We finally suggest several opportunities for further studies that may lead to novel applications in chemical and electromagnetic sensing, energy conversion, optoelectronics, quantum control, and quantum technology.

A mixture approach to the acoustic properties of a macroscopically inhomogeneous porous aluminum in the equivalent fluid approximation.

The acoustic properties of an air-saturated macroscopically inhomogeneous aluminum foam in the equivalent fluid approximation are studied. A reference sample built by forcing a highly compressible melamine foam with conical shape inside a constant diameter rigid tube is studied first. In this process, a radial compression varying with depth is applied. With the help of an assumption on the compressed pore geometry, properties of the reference sample can be modelled everywhere in the thickness and it is possible to use the classical transfer matrix method as theoretical reference. In the mixture approach, the material is viewed as a mixture of two known materials placed in a patchwork configuration and with proportions of each varying with depth. The properties are derived from the use of a mixing law. For the reference sample, the classical transfer matrix method is used to validate the experimental results. These results are used to validate the mixture approach. The mixture approach is then used to characterize a porous aluminium for which only the properties of the external faces are known. A porosity profile is needed and is obtained from the simulated annealing optimization process.

Magnetization non-rational quasi-plateau and spatially modulated spin order in the model of the single-chain magnet, [{(CuL)2Dy}{Mo(CN)8}]·2CH3CN·H2O

Using the exact solution in terms of the generalized classical transfer matrix method, we present a detailed analysis of the magnetic properties and ground-state structure of the simplified model of the single-chain magnet, trimetallic coordination polymer compound, , in which L2− is N,N-propylenebis(3-methoxysalicylideneiminato). Due to the presence of highly anisotropic Dy3+ ions, this material is a unique example of the one-dimensional magnets with Ising and Heisenberg bonds, allowing exact statistical-mechanical treatment. We found two zero-temperature ground states corresponding to different parts of the magnetization curve of the material. The zero-field ground state is shown to be an antiferromagnetic configuration with spatial modulation of the local Dy3+ (which is proven to posses well-defined Ising-like properties due to the large anisotropy of g-factors) and composite spin of the quantum spin trimer Cu-Mo-Cu in the form “up”-“down”-“down”-“up”. Another important feature of this compound is the appearance of the quasi-plateau at a non-rational value of magnetization due to the difference of the g-factors of the Cu- and Mo-ion in quantum spin trimers. The quasi-plateau is a nearly horizontal part of the magnetization curve where the corresponding zero-temperature ground state of the chain demonstrates slow, but monotonous dependence of the magnetization on the external magnetic field, while the z-projection of the total spin, , is constant.

Transfer matrix method applied to the parallel assembly of sound absorbing materials.

The transfer matrix method (TMM) is used conventionally to predict the acoustic properties of laterally infinite homogeneous layers assembled in series to form a multilayer. In this work, a parallel assembly process of transfer matrices is used to model heterogeneous materials such as patchworks, acoustic mosaics, or a collection of acoustic elements in parallel. In this method, it is assumed that each parallel element can be modeled by a 2 × 2 transfer matrix, and no diffusion exists between elements. The resulting transfer matrix of the parallel assembly is also a 2 × 2 matrix that can be assembled in series with the classical TMM. The method is validated by comparison with finite element (FE) simulations and acoustical tube measurements on different parallel/series configurations at normal and oblique incidence. The comparisons are in terms of sound absorption coefficient and transmission loss on experimental and simulated data and published data, notably published data on a parallel array of resonators. From these comparisons, the limitations of the method are discussed. Finally, applications to three-dimensional geometries are studied, where the geometries are discretized as in a FE concept. Compared to FE simulations, the extended TMM yields similar results with a trivial computation time.

Scattering of acoustic waves by macroscopically inhomogeneous poroelastic tubes

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. For planar configurations, the wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently: first in the case of rigid frame inhomogeneous porous materials and then in the case of inhomogeneous poroelastic materials in the framework of Biot's theory. This paper focuses on the solution of the full wave equation in cylindrical coordinates for poroelastic tubes in which the acoustic and elastic properties of the poroelastic tube vary in the radial direction. The reflection coefficient is obtained numerically using the state vector (or the so-called Stroh) formalism and Peano series. This coefficient can then be used to straightforwardly calculate the scattered field. To validate the method of resolution, results obtained by the present method are compared to those calculated by the classical transfer matrix method in the case of a two-layer poroelastic tube. As an example, a long bone excited in the sagittal plane is considered. Finally, a discussion is given of ultrasonic time domain scattered field for various inhomogeneity profiles, which could lead to the prospect of long bone characterization.

Propagation of acoustic waves in a one-dimensional macroscopically inhomogeneous poroelastic material

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. The wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently in the case of rigid frame inhomogeneous porous materials. This paper focuses on the solution of the full wave equation in which the acoustic and the elastic properties of the poroelastic material vary in one-dimension. The reflection coefficient of a one-dimensional macroscopically inhomogeneous porous material on a rigid backing is obtained numerically using the state vector (or the so-called Stroh) formalism and Peano series. This coefficient can then be used to straightforwardly calculate the scattered field. To validate the method of resolution, results obtained by the present method are compared to those calculated by the classical transfer matrix method at both normal and oblique incidence and to experimental measurements at normal incidence for a known two-layers porous material, considered as a single inhomogeneous layer. Finally, discussion about the absorption coefficient for various inhomogeneity profiles gives further perspectives.

In situ growth monitoring of AlGaN/GaN distributed Bragg reflectors at 530 nm using a 633 nm laser

The metal-organic chemical vapor deposition (MOCVD) growth of AlGaN/GaN distributed Bragg reflectors (DBR) with a reflection peak at 530 nm was in situ monitored using 633 nm laser reflectometry. Evolutions of in situ reflected reflectivity for different kinds of AlGaN/GaN DBR were simulated by the classical transfer matrix method. Two DBR samples, which have the same parameters as the simulated structures, were grown by MOCVD. The simulated and experimental results show that it is possible to evaluate the DBR parameters from the envelope shape of the in situ reflectivity spectrum. With the help of the 633 nm laser reflectometry, a DBR light emitting diode (LED) was grown. The room temperature photoluminescence spectra show that the reflection peak of the DBR in the LED is within the design region.

Two-dimensional dynamic simulations of DFB lasers and MOPAs

Due to the lateral structure of Distributed Feedback lasers (DFB) and Master Oscillator Power Amplifiers (MOPAs), the one-dimensional classical Transfer Matrix Method (TMM) as well as similar one-dimensional methods do not lead to satisfactory simulation results. Therefore, we present a two-dimensional simulation technique based on Trigonometric Finite Wave Elements (TFWE) — a generalization of the TMM in two or three dimensions — that solves the time-dependent wave equation. By coupling the wave equation with a temperature and a drift-diffusion model, we can simulate the time-dynamic behavior of DFB lasers and MOPAs. Furthermore, by Fourier transformation, we can investigate which modes and which frequencies appear. In this way, the influence of the injected current and of the stripe width on the resulting modes and on the output power can be analyzed in detail.

Acoustic wave propagation and internal fields in rigid frame macroscopically inhomogeneous porous media

A wave propagation model in macroscopically inhomogeneous porous media is derived from the alternative Biot’s theory of 1962. As a first application, the wave equation is reduced and solved in the case of rigid frame inhomogeneous porous materials. The pressure field, as well as the reflection and transmission coefficients, are obtained numerically using a wave splitting and “transmission” Green’s functions approach (WS-TGF). To validate both the wave equation and the method of resolution at normal and oblique incidence, results obtained by the WS-TGF method are compared to those calculated by the classical transfer matrix method and to experimental measurements for a known two-layered porous material, considered as a single inhomogeneous layer. Discussions are then given of the reflection and transmission coefficients for various inhomogeneity profiles as well as of the internal pressure field.

Acoustic wave propagation in a macroscopically inhomogeneous porous medium saturated by a fluid

The equations of motion in a macroscopically inhomogeneous porous medium saturated by a fluid are derived. As a verification of their validity, these equations are reduced and solved for rigid frame porous systems. The reflection and transmission coefficients and their corresponding time signals are calculated numerically using wave splitting Green’s function approach for a two-layer porous system considered as one single porous layer with a sudden change in physical properties. The results are compared to experimental results and to those of the classical transfer matrix method for materials saturated by air in the ultrasonic frequency range.

Eigenfrequency analysis of cable structures with inclined cables

The approximate eigenfrequencies for the in-plane vibrations of a cable structure consisting of inclined cables, together with point masses at various points were computed. It was discovered that the classical transfer matrix method was inadequate for this task, and hence the larger exterior matrices were used to determine the eigenfrequency equation. Then predictions of the dynamics of the general cable structure based on the asymptotic estimates of the exterior matrices were made.

Stable scattering-matrix method for surface acoustic waves in piezoelectric multilayers

A scattering matrix approach is proposed to avoid numerical instabilities arising with the classical transfer matrix method when analyzing the propagation of plane surface acoustic waves in piezoelectric multilayers. The method is stable whatever the thickness of the layers, and the frequency or the slowness of the waves. The computation of the Green’s function and of the effective permittivity of the multilayer is outlined. In addition, the method can be easily extended to the case of interface acoustic waves.