A uniform approximation for the eigenmodes of a planar dielectric waveguide

This paper derives a new approximation for the eigenmodes of a planar waveguide. The approximation is uniformly valid at both the high and low frequency regions of the dispersion relation. It is shown that a Pade approximation of the frequency equation leads to very accurate solutions. The new approximate solution is used to compute the frequency spectrum and the results compared with the exact analytical solution. The solutions presented here are ideal for analytically studying transient wave fields by means of modal summation.     

Modal theory for the two-frequency mutual coherence function in random media: beam waves

Pulse propagation in a random medium is mainly determined by the two-frequency mutual coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.     

Modal theory for the two-frequency mutual coherence function in random media: point source

The recently introduced modal expansion representation for the two-frequency mutual coherence function is applied here to the solution of a point-source field in a random medium. This approach reduces the solution for any structure function to an eigenvalue problem for an ordinary differential equation. For the initial point source it is shown here that the modal expansion yields a result similar to that for the initial plane wave, modified by a spherical free-space phase which contains a weighted coordinate that does not interact with the medium. Having established these general characteristics, special attention is paid to power-law media and, in particular, to a quadratic medium, for which a new exact solution is derived. Via a collective summation of this new modal solution, we rederive the alternative exact solution which exists in the literature. We also discuss the new parameterization implied by the new modal solution.     

Design of shaped piezoelectric modal sensors for cantilever beams with intermediate support by using differential transformation method

In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a cantilever beam with intermediate support is proposed by using the differential transformation method (DTM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function. By using boundary condition and continuity condition equations at intermediate support, the closed-form series solution of the second spatial derivative of the mode shapes can be determined based on DTM. Then the shapes of the designed modal sensors are obtained. Finally, numerical examples are given to demonstrate the feasibility of the proposed modal sensors for the cantilever beam with intermediate support.     

Vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices

This work provides a general formulation to solve vibration problems for continuous systems with damping effects, including modal, transient, harmonic and spectrum response analyses. In modal analysis, the system eigenvalues and corresponding eigenfunctions can be determined. The orthogonal relations of eigenfunctions are shown. For transient, harmonic and spectrum analyses, the generic force/actuator functions and response/sensing operators are introduced, respectively, and used to derive the system response. The time domain response is obtained for transient analysis, the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis. The solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force/actuator and response/sensor mode shape functions. In particular, one-dimensional beam and two-dimensional plate vibration analyses are illustrated by following the developed generic formulation. This work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well.     

Implementation of low-kurtosis pseudo-random excitations to compensate for the effects of nonlinearity on damping estimation by the half-power method

Pseudo-random excitation with low crest factor is less likely to force a structure under test into nonlinear behavior, which should be avoided, or at least minimized, in the practice of experimental modal analysis. However, simply cutting high peaks and removing them from the excitation time history is not an option because such clipping of the signal introduces frequency distortions of the amplitude spectrum. A better approach is to manipulate phases of the harmonics before generating the time history instead of clipping it afterwards. To do so a new parameter, kurtosis, is used in this paper to characterize the high peak behavior of pseudo-random excitations. An analytical solution is obtained for how the phases should be selected in order to reduce kurtosis and make modal testing excitations smoother with less extreme peaks. This solution was implemented for evaluation of the damping ratio of a SDOF system by the half-power method in the presence of an additional cubic term in the equation of motion. The system response obtained by numerical integration was treated as modal analysis data and the result is that the kurtosis-optimized excitation has compensated for the effect of nonlinearity and allowed to identify the damping ratio with good precision whereas an ordinary Gaussian excitation with randomized phases caused an error of 75 percent. Comparison with the numerical crest factor minimization by time-frequency-domain swapping has been made and experimental results from a modal testing rig with a realistic turbine blade are also presented in the paper.     

Modeling of axially moving wire with damping: Eigenfunctions, orthogonality and applications in slurry wiresaws

The classical modal analysis is applied to derive the analytical solution and to obtain the free vibration response of damped axially moving wire in this paper. The corresponding eigenvalues, eigenfunctions, and orthogonal relationship are presented. The orthogonality property and closed-form solution of free vibration response with damping are the main contributions of this study. In addition, the analytical modal analysis, with damping factor removed, shows agreement with those in existing research literature of moving wire without damping. The specific relevance of this general solution is discussed with respect to the moving wire in a slurry wiresaw. The theoretical definition of the damping factor of the slurry wiresaw system is also provided.     

Design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using Adomian decomposition method

The theory for designing distributed piezoelectric modal sensors is well established for beam structures. However, the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions (i.e., either clamped, free, simply supported or sliding). In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed, using the Adomian decomposition method (ADM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function. Based on the ADM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then the shapes of the designed modal sensors are obtained. Finally, some numerical examples are given to demonstrate the feasibility of the proposed modal sensors. It is shown that, for classical boundary conditions, the shapes of the modal sensors based on the ADM agree well with analytical and numerical results given in the literature. For general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another. The modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band.     

Free vibration analysis of circular cylindrical shells

A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders' shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTP). The numerical results obtained are compared with finite element method solutions.     

有球差多色高斯光束衍射引起的光谱开关

从空间频率域交叉谱密度函数的传输公式出发，推导出有球差多色高斯光束被硬边光阑衍射 后轴上光谱的解析公式，无像差的结果作为本公式的特例.对所得公式作了详细的数值计算 和物理分析.结果表明，球差高斯光束被硬边光阑衍射后近场的光谱与无像差时比较 有蓝移、红移，也会出现光谱开关.着重分析了球差对光谱开关的影响. 关键词： 现代光学 光阑衍射 光谱开关 球差 多色高斯光束     

Vibrations of rectangular plates reinforced by any number of beams of arbitrary lengths and placement angles

This paper presents an analytical method for the vibration analysis of plates reinforced by any number of beams of arbitrary lengths and placement angles. Both the plate and stiffening beams are generally modeled as three-dimensional (3-D) structures having six displacement components at a point, and the coupling at an interface is generically described by a set of distributed elastic springs. Each of the displacement functions is here invariably expressed as a modified Fourier series, which consists of a standard Fourier cosine series plus several supplementary series/functions used to ensure and improve uniform convergence of the series representation. Unlike most existing techniques, the current method offers a unified solution to the vibration problems for a wide spectrum of stiffened plates, regardless of their boundary conditions, coupling conditions, and reinforcement configurations. Several numerical examples are presented to validate the methodology and demonstrate the effect on modal parameters for a stiffened plate with various boundary conditions, coupling conditions, and reinforcement configurations.     

Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multi-point connections

Modal substructuring or component mode synthesis (CMS) has been standard practice for many decades in the analytical realm, yet a number of significant difficulties have been encountered when attempting to combine experimentally derived modal models with analytical ones or when predicting the effect of structural modifications using experimental measurements. This work presents a new method that removes the effects of a flexible fixture from an experimentally obtained modal model. It can be viewed as an extension to the approach where rigid masses are removed from a structure. The approach presented here improves the modal basis of the substructure, so that it can be used to more accurately estimate the modal parameters of the built-up system. New types of constraints are also presented, which constrain the modal degrees of freedom of the substructures, avoiding the need to estimate the connection point displacements and rotations. These constraints together with the use of a flexible fixture enable a new approach for joining structures, especially those with statically indeterminate multi-point connections, such as two circular flanges that are joined by many more bolts than required to enforce compatibility if the substructures were rigid. Fixture design is discussed, one objective of which is to achieve a mass-loaded boundary condition that exercises the substructure at the connection point as it is in the built up system. The proposed approach is demonstrated with two examples using experimental measurements from laboratory systems. The first is a simple problem of joining two beams of differing lengths, while the second consists of a three-dimensional structure comprising a circular plate that is bolted at eight locations to a flange on a cylindrical structure. In both cases frequency response functions predicted by the substructuring methods agree well with those of the actual coupled structures over a significant range of frequencies.     

On the transmission coefficient for the double delta-function potential embedded in a crystalline lattice

The solution of the Schrödinger–Wannier equation is examined for the potential that is defined as a sum of the two delta functions of unequal strengths. The analytical expression for the transmission coefficient is derived from the solution. The transmission coefficient is shown to exhibit maxima and minima. Moreover, it is proved that the transmission coefficient in its maxima is larger and in its minima is smaller than the transmission coefficient for the corresponding single delta-function potential. Some differences between the transmission coefficient of particles with the quadratic energy spectrum and the transmission coefficient of particles with the Kane energy spectrum are presented.     

New method for solving fractional partial integro-differential equations by combination of Laplace transform and resolvent kernel method

In this article, we obtained the approximate solution for a new class of Time-Fractional Partial Integro-Differential Equation (TFPIDE) of the Caputo-Volterra type in which the integral is not limited to the convolution type. This new class of TFPIDE is distinct from the common problem with the convolution integral kernel. The general expression of the analytical solution for this special type of TFPIDE was derived using a combination of Laplace transform and the resolvent kernel method. In the process, Laplace transform will transform the equation into a second kind Volterra integral equation in terms of the transformed function. Two main problems in deriving the approximate analytical solutions were identified as Case I and Case II problems. To obtain the approximate solutions for Case I and Case II problems, numerical methods were designed based on approximation of the resolvent kernel with truncated Neumann series as well as approximation of the Laplace transform based on truncated Taylor series. Several numerical examples are presented to indicate the plausibility, mechanism and performance of the proposed methods.     

Acoustic characteristics analysis of slender cavity system with arbitrary impedance end conditions

A modeling method for the dynamic characteristics analysis of a slender acoustical cavity with impedance end conditions is established. In order to satisfy the continuity requirement at impedance ends for the first order differential of sound pressure, field function is constructed as the standard Fourier series supplemented by boundary smoothed auxiliary polynomials. System characteristic equation is derived by solving the governing differential equation and impedance acoustic boundary of slender acoustical cavity system simultaneously,relevant acoustical modal information is obtained via the state space solution procedure. In numerical simulation, various acoustic variables, such as acoustical modal frequency, sound pressure modal shape, sound pressure response and the particle velocity, are presented for the slender acoustical cavity system with different boundary conditions and compared with those results in the existing literature. The correctness and effectiveness of the proposed method are then fully validated.     

双环形Hulthén势束缚态的近似解析解

构造了双环形Hulthén势, 用指数函数近似表示任意分波的离心项, 运用函数分析法讨论双环型Hulthén势Schrödinger方程的束缚态解. 归一化的角向波函数和径向波函数用超几何多项式表示, 给出了束缚态能谱, 体系的束缚态的能谱方程和波函数与量子数和势参数有关. 中心势场和单环形势场角向波函数及 Hulthén势束缚态能谱是本文双环形Hulthén势的特例. 关键词： 双环形Hulthén势 任意分波 近似解析解 束缚态     

考虑任意阻抗壁面条件管腔结构声场特性分析

针对任意阻抗壁面条件一维管腔声学系统建模,对系统动力学特性进行预报。为了满足阻抗边界条件对声压一阶导数连续性要求,管腔声压函数通过在标准傅里叶级数端点位置引入边界光滑辅助多项式进行构建。结合壁面阻抗声学边界和管腔声学Helmholtz控制微分方程得到强形式标准特征值问题,获得相应的声学模态信息。在数值仿真中,通过算例给出各种边界条件下管腔声学模态频率、声压振型、声压和质点振速频率响应曲线,与现有文献中相关结果进行对比,充分验证了本文求解方法的正确性和有效性,证明该方法可对任意阻抗壁面条件管腔系统声学特性进行准确预报。     

Quantum theory of synchronously pumped type I optical parametric oscillators: characterization of the squeezed supermodes

Quantum models for synchronously pumped type I optical parametric oscillators (SPOPO) are presented. The study of the dynamics of SPOPOs, which typically involves millions of coupled signal longitudinal modes, is significantly simplified when one considers the “supermodes", which are independent linear superpositions of all the signal modes diagonalizing the parametric interaction. In terms of these supermodes the SPOPO dynamics becomes that of about a hundred of independent, single mode degenerate OPOs, each of them being a squeezer. One derives a general expression for the squeezing spectrum measured in a balanced homodyne detection experiment, valid for any temporal shape of the local oscillator. Realistic cases are then studied using both analytical and numerical methods: the oscillation threshold is derived, and the spectral and temporal shapes of the squeezed supermodes are characterized.     

Modal coupling in acoustic waveguides: planar discontinuities

A new derivation for a scattering matrix for reflection and transmission of higher order modes at the planar junction of two waveguides is presented. The derivation is extended to include finite junction wall impedance and offset waveguides. The resulting matrix equations are analyzed and the physical significance of the matrices is explained. As an example of the theory, analytical expressions for the coupling coefficients at a size change in a rectangular duct are developed and the resulting reflection and transmission coefficients are computed. The results should be of interest to the HVAC noise control community. The paper also shows the effects of modal truncation on the accuracy and convergence of the solution. It is shown that the proper selection of the ratio of the number of modes on either side of the discontinuity is related to the ratio of the characteristic sizes of the waveguides. Finally it is shown that at least one higher mode should be included for reasonable accuracy in the computation of plane wave reflection and transmission coefficients except at the very lowest frequencies.     

双包层椭圆光波导解析解

解析求解了双包层椭圆光纤中的波动方程，得到了模式精确解有模式特征方程。对基模的特征方程进行了数值计算，给出了不同椭圆比下的归一化双折射和模间色散随归一化频率的变化关系曲线，并与高斯近似解的结果进行了比较。