Dynamic stiffness vibration analysis of an elastically connected three-beam system

An exact dynamic stiffness method is developed for predicting the free vibration characteristics of a three-beam system, which is composed of three non-identical uniform beams of equal length connected by innumerable coupling springs and dashpots. The Bernoulli-Euler beam theory is used to define the beams’ dynamic behaviors. The dynamic stiffness matrix is formulated from the general solutions of the basic governing differential equations of a three-beam element in damped free vibration. The derived dynamic stiffness matrix is then used in conjunction with the automated Muller root search algorithm to calculate the free vibration characteristics of the three-beam systems. The numerical results are obtained for two sets of the stiffnesses of springs and a large variety of interesting boundary conditions.     

Radiation of ultrasound into an anisotropic solid

A half-space, x₃ 0, of a transversely isotropic solid whose axis of symmetry makes an angle with the x₃-axis, is subjected to a spatially uniform time-harmonic distribution of normal surface tractions over a circular area of the plane x₃ = 0. The wave motion radiated into the half-space is investigated. Using an integral representation for the displacement components the problem is first reduced to a system of singular integral equations for the displacements on the surface x₃ = 0. This system is solved by the boundary element method over a truncated area, where use is made of recently derived simplified forms of the Green's functions. The results show the skewing of the beam as the angle between the axis of symmetry of the transversely isotropic solid and the normal to the surface of the solid is increased.     

Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions

The propagation of time-harmonic plane elastic waves in infinite elastic composite materials consisting of linear elastic matrix and rigid penny-shaped inclusions is investigated in this paper. The inclusions are allowed to translate and rotate in the matrix. First, the three-dimensional (3D) wave scattering problem by a single inclusion is reduced to a system of boundary integral equations for the stress jumps across the inclusion surfaces. A boundary element method (BEM) is developed for solving the boundary integral equations numerically. Far-field scattering amplitudes and complex wavenumbers are computed by using the stress jumps. Then the solution of the single scattering problem is applied to estimate the effective dynamic parameters of the composite materials containing randomly distributed inclusions of dilute concentration. Numerical results for the attenuation coefficient and the effective velocity of longitudinal and transverse waves in infinite elastic composites containing parallel and randomly oriented rigid penny-shaped inclusions of equal size and equal mass are presented and discussed. The effects of the wave frequency, the inclusion mass, the inclusion density, and the inclusion orientation or the direction of the wave incidence on the attenuation coefficient and the effective wave velocities are analysed. The results presented in this paper are compared with the available analytical results in the low-frequency range.     

Albedo problem for finite plane-parallel medium

The problem of radiation transfer through a scattering and absorbing finite plane-parallel medium is solved using an efficient and accurate method of analysis which utilizes trial functions based on Case's eigenvalues plus a linear combination of exponential integral functions. The proposed trial functions are used on the integral equation reducing it to a system of algebraic equations to be solved for the expansion coefficients which are used to calculate some interesting physical quantities such as the angular radiation intensity and the reflection and the transmission coefficients. Numerical results are obtained for two different external incidence on the left boundary, x=0. The results are compared with the exact results and with those calculated by the Pomraning-Eddington variational method.     

An analytical technique for calculation of distributed resistance in a rectangular domain

A Green's function approach is used to obtain an integral representation for the electric potential in a multi-terminal distributed resistive structure. This integral representation for the potential is then used to formulate a coupled system of Fredholm integral equations of the first kind, in which the normal components of current density over the different terminal contacts are the unknowns. A procedure for solving this system of equations is presented and the obtained results for the unknown normal components of the current density are used to express the electric potential and current density vector at any point in the domain of the distributed resistive structure. The indefinite admittance matrix relating the terminal currents to the applied terminal voltages, is a by-product of the solution process. Explicit expressions for the Green's function and complete orthonormal sets of functions which are required to apply the solution to a rectangular domain are given. Applications to three-terminal structures are considered in order to illustrate the method.     

On dynamic plane elasticity problems of 2D quasicrystals

In this Letter, the dynamic plane elasticity problems of 2D quasicrystals is considered. By use of the Fourier transform and matrix transformations the system is reduced to uncoupled ordinary differential equations. Fourier images of Green's functions for dynamic plane elasticity problems of 2D dodecagonal, pentagonal and decagonal quasicrystals are obtained explicitly by the suggested method.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

SH-wave propagation and resonance phenomena in a periodically layered composite structure with a crack

This paper presents a dynamic analysis of time-harmonic plane SH-waves propagating in periodically multilayered elastic composites with a strip-like crack. The total wave field in the multilayered elastic structure is described as a sum of incident wave field modeled by the transfer matrix method and the scattered wave field governed by an integral representation containing the crack-opening-displacement. The integral equation derived from the boundary conditions on the crack-faces is solved numerically by a Galerkin method. The paper focuses on resonant and non-resonant regimes of anti-plane wave motion in a stack of elastic layers weakened by a single strip-like crack and wave localization in the vicinity of the crack. The scattered extra displacement induced by the presence of the crack is investigated in detail for both situations of high and low contrast in material properties. Numerical results for the average crack-opening-displacement, the transmission coefficient, the stress intensity factor and the average energy flow are presented and discussed to reveal wave resonance and localization phenomena within the band-gaps and the pass-bands.     

Vibration analysis of liquid-filled pipelines with elastic constraints

In this paper, a transfer matrix method (TMM) in frequency domain considering fluid-structure interaction of liquid-filled pipelines with elastic constraints is proposed. The time-domain equations considering fluid-structure interaction, are transformed into frequency domain by Laplace transformation, and then twelve fourth-order ordinary differential equations and two second-order ordinary differential equations are deduced from the frequency-domain equations. The results of the fourteen frequency-domain equations are assembled into a transfer matrix, which represents the motion of a single pipe section. Combined with point matrices that describe specified boundary conditions, an overall transfer matrix for liquid-filled pipeline system can be assembled. Using the method, all the pipeline with no and rigid constraints can be easily calculated by simply setting the stiffness of the restraining springs from zero to a large number. Taking into account the longitudinal vibration, transverse vibration and torsional vibration, the proposed method can be used to analyze the pipelines with bends. Several numerical examples with different constraints are presented here to illustrate the application of the proposed method. The results are validated by measured and simulation data. Through the numerical examples, it is shown that the proposed method is efficient.     

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.     

基于新型混合场积分方程的半空间三维均匀介质目标电磁散射分析

利用介质半空间格林函数,将一种混合场积分方程(JMCFIE)推广应用到有耗半空间的均匀介质的散射分析中,以减少离散矩阵方程迭代求解时所需的迭代步数,提高矩量法(MoM)对半空间介质目标散射分析的求解效率.数值算例验证了方法的准确性和有效性.     

On the formal expansion of the possibilities of the method of integral transformations in investigating linear distributed systems with constant parameters

The Green's functions for a series of boundary-value problems are formulated in transforms with respect to t by means of matrix functions that are introduced and allow the scalar dependence on the weighted differences of the coordinate argument to be isolated in them. It was possible in general form to carry out the procedure of the method of integral transformations in solving the integral equations for the function that generates periodic motion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 3, pp. 323–331, March, 1972.     

Mean field approximation versus exact treatment of collisions in few-body systems

A variational principle for calculating matrix elements of the full resolvent operator for a many-body system is studied. Its mean field approximation results in nonlinear equations of Hartree (-Fock) type, with initial and final channel wave functions as driving terms. The mean field equations will in general have many solutions whereas the exact problem being linear, has a unique solution. In a schematic model with separable forces the mean field equations are analytically soluble, and for the exact problem the resulting integral equations are solved numerically. Comparing exact and mean field results over a wide range of system parameters, the mean field approach proves to be a very reliable approximation, which is not plagued by the notorious problem of defining asymptotic channels in the time-dependent mean field method.     

Biorthogonality and radiative transfer in finite slab atmospheres

It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation.The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and non-linear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.     

Calculation of electromechanical coupling coefficient of Lamb waves in multilayered plates

Two methods have been always used to calculate the electromechanical coupling coefficient of a Lamb wave in a multilayered plate: one is an approximate method using the acoustic velocity difference under different electric boundary conditions and the other is the Green's function method. The Green's function method is more accurate but more complicated, because an 8N-order matrix is used for calculating the electromechanical coupling coefficient of the Lamb wave in an N-layered plate, which induces great computation loads and some calculation deviations. In this paper, a transfer matrix method is used for calculating the electromechanical coupling coefficient of Lamb waves in a multilayered plate, in which only an 8-order matrix is needed regardless of the number of layers of the plate. The results show that the transfer matrix method can obtain the same accuracy as those by the Green's function method, but the computation load and deviation are greatly decreased by avoiding the use of a high order matrix used in the Green's function method.     

Exact wave field simulation for finite-volume scattering problems

An exact boundary condition is presented for scattering problems involving spatially limited perturbations of arbitrary magnitude to a background model in generally inhomogeneous acoustic media. The boundary condition decouples the wave propagation on a perturbed domain while maintaining all interactions with the background model, thus eliminating the need to regenerate the wave field response on the full model. The method, which is explicit, relies on a Kirchhoff-type integral extrapolation to update the boundary condition at every time step of the simulation. The Green's functions required for extrapolation through the background model are computed efficiently using wave field interferometry.     

任意边界的Stokes内流问题的数值计算

本文采用一种格林函数方法^[1],研究了包括任意形状的外边界和流场中任意形状的物体的Stokes内流问题。将方程化成边界积分,从而降低一个维度。数值地求解边界面上的应力分量,再进一步算出流动区域的速度场和压力场,各种不同形状边界的算例,都得到了满意的结果。在处理任意边界内的任意形状物体的绕流时,本方法较其它方法有着明显的优点。     

夸克-反夸克束缚态的B-S相互作用核的封闭表示式(英文)

根据从 QCD生成泛函所建立的夸克和反夸克的传播子、四点格林函数及其它类型的格林函数所满足的运动方程 ,推导出了夸克 -反夸克束缚态的 Bethe- Salpeter方程中相互作用核的明显且封闭的表示式 ,给出了这个表示式的未重整化和重整化了的形式 .这个表示式不仅易于进行微扰计算 ,而且适于进行非微扰的计算 ,特别是它提供了求解夸克禁闭问题一个恰当的理论出发点.The interaction kernel in the Bethe-Salpeter equation for quark-antiquark bound states is derived from the Bethe-Salpeter equations satisfied by the quark-antiquark four-point Green s function. The latter equations are established based on the equations of motion obeyed by the quark and antiquark propagators, the four-point Green s function and some other kinds of Green s functions which follow directly from the QCD generating function. The Bethe-Salpeter kernel derived is an exact...     

Localized vibrations of point defects in body-centred cubic lattices

The paper gives an exact calculation of the localized frequencies of substitutional defects in a body-centred cubic lattice by the method of Green's functions and compares it with the approximate calculation carried out after [14]. The exact calculation is based on newly computed Green's functions of a b.c.c. lattice [18]. It is shown how by means of group theory the symmetry of the system can be used both in an approximate and in the exact calculation. Some symmetry relations between Green's functions, which limit the number of functions necessary for numerical calculations, are derived.     

Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.