Calculation Of Eigenfrequencies Research Articles

Lattice-dynamics study of Raman-active modes in LaTiO3

The lattice dynamics plays an important role in the determining the orbital and magnetic ordering phenomena in rare-earth (R) RTiO3 titanates. In the present study, the assignment and vibrational character of Ag and B2g-symmetry phonons observed in the Raman spectra of LaTiO3 crystal is verified by shell-model calculations of eigenfrequencies, relative intensities, and eigenvector components of the normal modes. The eigenvector pattern of the Ag mode located around the most pronounced central mode at 296cm−1 is turned out to be in full agreement with the La–O covalency in LaTiO3, where the concomitant GdFeO3-type distortion is decisive. The GdFeO3-type mode is located in the close proximity to the Jahn-Teller (JT) tetragonal E-type Qε mode. An interplay between the R–O covalency and JT effects can determine the subtle conditions for the electronic ground state of the Ti3+ ion and associated orbital and magnetic ordering phenomena in RTiO3 titanates.

A Mutual Coupling Theory for Arbitrarily Distributed Trimeric Localized Surface Plasmonic Particles

Coupled mode theory (CMT) is widely used to analyze the coupling relationship of plasmons and metasurfaces. However, some details involving coupled structures remain obscure. Herein, the coefficients in CMT are properly defined to solve the problems of inaccurate fitting from two particles to three or more. The proposed improved coupled mode theory (i‐CMT) can perfectly express both the amplitude and phase of transmission curve of dimeric and trimeric mutual coupled systems. A surface plasmonic structure containing multiple metallic spiral structures is fabricated to experimentally verify the accuracy of the i‐CMT. The experimental results are consistent with the theoretical and simulated values, proving the feasibility of using the i‐CMT to analyze and design asymmetric coupled devices. Moreover, the application of the i‐CMT confirms the presence of additional coupling terms in three or more coupled objects, making the calculation of eigenfrequency more accurate.

Surface effects and turbulent pressure

The application of the full potential of stellar seismology is made difficult by the improper modelling of the upper-most layers of solar-like stars and their influence on the modelled frequencies. Our knowledge of these so-called ‘surface effects’ has improved thanks to the use of 3D hydrodynamical simulations, however, the calculation of eigenfrequencies relies on empirical models for the description of the Lagrangian perturbation of turbulent pressure, namely: the reduced-Γ1 model (RGM) and the gas-Γ1 model (GGM). Starting from the fully compressible turbulence equations, we derived both the GGM and RGM models by using a closure to model the flux of turbulent kinetic energy. We find that both models originate from two terms: the source of turbulent pressure due to compression produced by the oscillations and the divergence of the flux of turbulent pressure. We also demonstrate that they are both compatible with the adiabatic approximation and, additionally, that they imply a number of questionable assumptions, mainly with respect to mode physics. Among other hypotheses, it is necessary to neglect the Lagrangian perturbation of the dissipation of turbulent kinetic energy into heat and the Lagrangian perturbation of buoyancy work.

Calculation of eigen frequencies of systems with damping

This paper considers a problem of detecting all resonance frequencies during the design phase for complex technical objects made of sub-systems of various physical natures with the unknown point of application of a harmonic input. Consequently, it is required to choose a proper calculation method for eigen frequencies and eigen vectors for those systems. Results. Is to analyze mathematical methods for calculating eigen frequencies and eigen vectors of complex technical systems with sub-systems of different nature and give recommendations on selecting a proper method and its justification. Results. It is demonstrated that in order to calculate eigen frequencies of mechanical systems, a use of a system of Cauchy differential equations is appropriate because it allows avoiding limitations caused by a traditional mathematical model. It is shown that the eigen frequency and eigen vector analysis helps one to determine eigen frequencies in presence of dampening as well as the ratio of amplitude of oscillation on those frequencies. Practical significance. The proposed method for selecting calculation technique for eigen frequencies and eigen vectors helps one to determine eigen frequencies in presence of dampening as well as the ratio of amplitude of oscillation on those frequencies. This method is applicable to the nonlinear system analysis in arbitrary points of nonlinearity.

A method of effective potentials for calculating the frequency spectrum of eccentrically layered spherical cavity resonators

ABSTRACTA novel method for the calculation of eigenfrequencies of non-uniformly filled spherical cavity resonators is developed. The spectrum is determined through the introduction of effective “dynamic” potentials which encode the infill inhomogeneity. For angularly symmetric cavity, regardless of its radial non-uniformity, the set of modes is shown to be a superposition of TE and TM oscillations which can be described in terms of a single scalar function independently of each other. The violation of polar symmetry in the infill dielectric properties, the azimuthal symmetry being preserved, suppresses all azimuthally non-uniform modes of TM oscillations. In the absence of angular symmetry of both electric and magnetic properties of the infill, only azimuthally uniform distribution of both TM and TE fields is expected to occur in the resonator. The method is quite efficient for computational complex algorithms for solving different spectral problems, including those for studying the chaotic properties of systems' spectra.

Effects of meshing density of 1D structural members with non-uniform cross-section along the length on the calculation of eigenfrequencies

Density of division in finite element method does not affect only the accuracy of calculation, but also the necessary calculation time. This is directly influenced by the power of the used hardware, the efficiency of the algorithm used to assemble global stiffness and mass matrices and finally, by the method used to find eigenvalues of matrices for determination of eigenfrequencies. In engineering practice, when commercially available software is used, it is necessary to look for the optimum between the accuracy of the calculation and the length of the calculation. This paper deals with solution of eigenfrequencies of 1D elements with nonuniform cross section using Python 3.7.4 with libraries "numpy 1.18.1" for finding eigenvalues of matrices and "scipy 1.4.1" for finding solution for system of nonlinear equations.

Waveguide analysis for ultrasonic medical instruments (comparison of different methods)

Transition matrices are derived on the basis of some analytical solutions for a number of waveguides. Initial parameters method is used to analyze the waveguide longitudinal oscillations. This paper contains a comparative analysis of different types of concentrators. The waveguide design scheme presents a straight bar of variable cross-section with different boundary conditions. For given concentrator arrangements, the amplitude axial displacements and longitudinal forces in the ultrasonic medical instrument (UMI) are calculated for a certain excitation frequency. Numerical method for calculation of eigenfrequencies and eigenmodes of the ultrasonic medical instrument (UMI) waveguide is presented. Analysis of the cantilever straight-line concentrator based on the methods of transition matrices and numerical integration by the Runge-Kutta method are under comparison. Sufficient agreement of the results is demonstrated.

Calculation of eigen frequencies of a rope moving in longitudinal direction

Calculation of eigen frequencies of a rope moving in longitudinal direction

An Efficient Approach for the Calculation of Eigenfrequencies and Mode Shapes of Tapered and Axially Functionally Graded Rayleigh Beams

An Efficient Approach for the Calculation of Eigenfrequencies and Mode Shapes of Tapered and Axially Functionally Graded Rayleigh Beams

Analytical estimates of eigenfrequencies, dispersion, and field distribution in whispering gallery resonators

The problem of accurate calculation of eigenfrequencies in resonators of complex geometry is not only fundamental but also has many practical applications. In particular, a possibility for calculating the eigenfrequencies and geometry dependent dispersion of whispering gallery modes is important for optimization of dielectric microresonator-based Kerr frequency combs. In this case, the required anomalous second-order dispersion may be controlled by means of small shape variations of the resonator. Unfortunately, all uniform approximations for the eigenfrequencies do not reach the required precision for this purpose. We propose new approximations for spheroids, quartics, and toroids with better precision, which also allow for the estimation of the second-order dispersion. We also obtain analytical expressions for field distribution in microresonators and investigate the possibility of achieving better approximations by combining analytical and numerical methods.

Approximate calculation of eigenfrequencies of biconical microwave cavities

Simplified expressions are obtained for the calculation of the eigenfrequencies of a biconical cavity with large apex angles with the use of two symmetrical points at the boundaries of overlapping partial regions in the form of a spherical sector. Using these expressions, the dependence of the cavity eigenfrequencies on the apex angles of the conical elements is studied, and the results are compared with those obtained by the classical finite-element method. The applicability range of the simplified expressions is identified, and recommendations on their use are given.

EIGENFREQUENCIES OF THE REINFORCED CONCRETE BEAMS – METHODS OF CALCULATIONS / GELŽBETONINIŲ SIJŲ TIKRINIAI DAŽNIAI – SKAIČIAVIMO METODAI

The paper presents method of calculation of eigenfrequencies of the cracked reinforced concrete beams including discreet model of crack. The described method is based on the stiff finite elements method. It was modified in such a way as to take into account local discontinuities – cracks. In addition, some theoretical studies as well as experimental tests of concrete mechanics based on discrete crack model were taken into consideration. The calculations were performed using the author's own numerical algorithm. Moreover, other calculation methods of dynamic reinforced concrete beams presented in standards and guidelines are discussed. Calculations performed by using different methods are compared with the results obtained in experimental tests. Santrauka Straipsnyje pristatomas įtrūkusių gelžbetonio sijų tikrinių dažnių skaičiavimo metodas, įskaitant diskretinį įtrukio modelį. Metodas pagrįstas standžių baigtinių elementų metodu. Jis buvo patobulintas siekiant įtraukti vietinius nevienalytiškumus – įtrūkius. Taip pat buvo įvertintos kai kurios betono mechanikos teorines studijos ir eksperimentiniai bandymai, pagrįsti diskrečiu įtrūkių modeliu. Skaičiavimai buvo atlikti taikant autorių sukurtą skaitmeninį algoritmą. Be to, pristatomi ir aptariami kiti dinaminiai gelžbetoninių sijų skaičiavimo metodai, pateikti standartuose ir gairėse. Skaičiavimai, atlikti taikant skirtingus metodus, palyginami su rezultatais, gautais eksperimentinių bandymų metu.

First-Principles Full-Vectorial Eigenfrequency Computations for Axially Symmetric Resonators

Starting from the time-harmonic Maxwell's equations in cylindrical coordinates, we derive and solve the finite-difference (FD) eigenvalue equations for determining vector modes of axially symmetric resonator structures such as disks, rings, spheres and toroids. Contrary to the most existing implementations, our FD scheme is readily adapted for both eigenmode and eigenfrequency calculations. An excellent match of the FD solutions with the analytically calculated mode indices of a microsphere resonator provides a numerical confirmation of the mode-solver accuracy. The comparison of the presented FD technique with the finite-element method highlights the relative strengths of both techniques and advances the FD mode-solver as an important tool for cylindrical resonator design.

Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions

In this work, we study the application of the Method of Fundamental Solutions (MFS) for the calculation of eigenfrequencies and eigenmodes in two and three-dimensional domains. We address some mathematical results about properties of the single layer operator related to the eigenfrequencies. Moreover, we propose algorithms for the distribution of the collocation and source points of the MFS in three-dimensional domains which is an extension of the choices considered by Alves and Antunes (CMC 2(2005), 251–266) for the two-dimensional case. Also the application of the Plane Waves Method is investigated. Several examples with Dirichlet and Neumann boundary conditions are considered to illustrate the performance of the proposed methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1525–1550, 2011

On the choice of source points in the method of fundamental solutions

The method of fundamental solutions (MFS) may be seen as one of the simplest methods for solving boundary value problems for some linear partial differential equations (PDEs). It is a meshfree method that may present remarkable results with a small computational effort. The meshfree feature is particularly attractive when we need to change the shape of the domain, which occurs, for instance, in shape optimization and inverse problems. The MFS may be viewed as a Trefftz method, where the approximations have the advantage of verifying the linear PDE, and therefore we may bound the inner error from the boundary error, in well-posed problems. A main counterpart for these global numerical methods, that avoid meshes, are the associated linear systems with dense and ill conditioned matrices. In these methods a sort of uncertainty principle occurs— we cannot get both accurate results and good conditioning—one of the two is lost. A specific feature of the MFS is some freedom in choosing the source points. This might lead to excellent results, but it may also lead to poor results, or even to impossible approximations. In this work we will discuss the choice of source points and propose a choice along the discrete normal direction (following [Alves CJS, Antunes PRS. The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput Mater Continua 2005;2(4):251–66]), with a possible local criterion to define the distance to the boundary. We will also address some extensions that connect the asymptotic MFS to other methods by choosing the sources on a circle/sphere far from the boundary. We also present a direct connection between the approximation based on radial basis functions (RBF) and the MFS approximation in a higher dimension. This increase in dimension was somehow already present in a previous work [Alves CJS, Chen CS. A new method of fundamental solutions applied to non-homogeneous elliptic problems. Adv Comput Math 2005;23:125–42], where the frequency was used as the extra dimension. The free parameters in RBF inverse multiquadrics 2D approximation correspond in fact to the source point distance to the boundary plane in a Laplace 3D setting. Some numerical simulations are presented to illustrate theoretical issues.

3-D eigenmode calculation of metallic nano-structures

Abstract. In the calculation of eigenfrequencies of 3-D metallic nanostructures occurs the challenge that the material parameters depend on the desired eigenfrequency. We propose a formulation where this leads to a polynomial eigenvalue problem which can be tackled by different solving strategies. A comparison between a Newton-type method and a Jacobi-Davidson algorithm is given.

Phononic properties of opals

We study the vibrational properties of opals consisting of silica spheres stacked periodically. To analyze their characteristic features (frequency gaps etc.) we start from the calculation of eigenfrequencies of a single sphere and see how they are modulated when spheres are combined to form a single layer and then multiple layers stacked periodically. The theoretical band structure is compared with the vibrational spectra measured experimentally.

Two-layered disc quasi-optical dielectric resonators: electrodynamics and application perspectives for complex permittivity measurements of lossy liquids

Electromagnetic properties of novel quasi-optical resonators are studied theoretically and experimentally. The resonators are a radially two-layered dielectric disc sandwiched between conducting endplates. The internal layer can be filled with air or lossy liquid. Whispering gallery modes are excited in such a resonator and the mode energy is concentrated near the inner side of the cylindrical surface of an external layer. The measurement data obtained in the Ka-band are compared with theoretical calculations of eigenfrequencies and quality factors of the Teflon resonator filled with water, ethyl alcohol, benzene and aqueous solutions of ethyl alcohol. A number of ‘anomalous’ properties of the resonator can be described using Maxwell equations. The experimental data on the complex permittivity of a binary mixture water–ethyl alcohol are compared with the values calculated in terms of Debye's function. An important feature of the proposed technique is that it holds promise for making first principle microwave measurements of the permittivity of lossy liquids.

Geometrical theory of whispering-gallery modes

Using a quasi-classical approach, rather precise analytical approximations for the eigenfrequencies of whispering-gallery modes (WGMs) in convex axisymmetric bodies may be found. We use the eikonal method to analyze the limits of precision of quasi-classical approximation using, as a practical example, a spheroidal dielectric cavity. The series obtained for the calculation of eigenfrequencies is compared with the known series for a dielectric sphere, and with numerical calculations. We show how geometrical interpretation allows expansion of the method on arbitrarily-shaped axisymmetric bodies.

Continuum limit of amorphous elastic bodies. III. Three-dimensional systems

Extending recent numerical studies on two dimensional amorphous bodies, we characterize the approach of elastic continuum limit in three dimensional (weakly polydisperse) Lennard-Jones systems. While performing a systematic finite-size analysis (for two different quench protocols) we investigate the non-affine displacement field under external strain, the linear response to an external delta force and the low-frequency harmonic eigenmodes and their density distribution. Qualitatively similar behavior is found as in two dimensions. We demonstrate that the classical elasticity description breaks down below an intermediate length scale $\xi$, which in our system is approximately 23 molecular sizes. This length characterizes the correlations of the non-affine displacement field, the self-averaging of external noise with distance from the source and gives the lower wave length bound for the applicability of the classical eigenfrequency calculations. We trace back the "Boson-peak" of the density of eigenfrequencies (obtained from the velocity auto-correlation function) to the inhomogeneities on wave lengths smaller than $\xi$.