On the forced response of waveguides using the wave and finite element method

The forced response of waveguides subjected to time harmonic loading is treated. The approach starts with the wave and finite element (WFE) method where a segment of the waveguide is modeled using traditional finite element methods. The mass and stiffness matrices of the segment are used to formulate an eigenvalue problem whose solution yields the wave properties of the waveguide. The WFE formulation is used to obtain the response of the waveguide to a convected harmonic pressure (CHP). Since the Fourier transform of the response to a general excitation is a linear combination of the responses to CHPs, the response to a general excitation can be obtained via an inverse Fourier transform process. This is evaluated analytically using contour integration and the residue theorem. Hence, the approach presented herein enables the response of a waveguide to general loading to be found by: (a) modeling a segment of the waveguide using finite element methods and post-processing it to obtain the wave characteristics, (b) using Fourier transform and contour integration to obtain the wave amplitudes and (c) using the wave amplitudes to find the response at any point in the waveguide. Numerical examples are presented.     

Calculating the forced response of cylinders and cylindrical shells using the wave and finite element method

The dynamic response of circular cylinders can be obtained analytically in very few (and simple) cases. For complicated (thick or anisotropic) circular cylinders, researchers often resort to the finite element (FE) method. This can lead to large models, especially at higher frequencies, which translates into high computational costs and memory requirements. In this paper, the response of axially homogenous circular cylinders (that can be arbitrarily complex through the thickness) is obtained using the wave and finite element (WFE) method. Here, the homogeneity of the cylinder around the circumference and along the axis are exploited to post-process the FE model of a small rectangular segment of the cylinder using periodic structure theory and obtain the wave characteristics of the cylinder. The full power of FE methods can be utilised to obtain the FE model of the small segment. Then, the forced response of the cylinder is posed as an inverse Fourier transform. However, since there are an integer number of wavelengths around the circumference of a closed circular cylinder, one of the integrals in the inverse Fourier transform becomes a simple summation, whereas the other can be resolved analytically using contour integration and the residue theorem. The result is a computationally efficient technique for obtaining the response to time harmonic, arbitrarily distributed loads of axially homogenous, circular cylinders with arbitrary complexity across the thickness.     

Far-field approximation for a point-excited anisotropic plate

An analytic approximation is derived for the far-field response of a generally anisotropic plate to a time-harmonic point force acting normal to the plate. This approximation quantifies the directivity of the flexural wave field that propagates away from the force, which is expected to be useful in the design and testing of anisotropic plates. Derivation of the approximation begins with a two-dimensional Fourier transform of the flexural equation of motion. Inversion to the spatial domain is accomplished by contour integration over the radial component of wave number followed by an application of the method of stationary phase to integration over the circumferential component of wave number. The resulting approximation resembles that of an isotropic plate but involves wave numbers, wave amplitudes, and phases that depend on propagation angle. Numerical results for a plate comprised of bonded layers of a graphite-epoxy material illustrate the accuracy of the method compared to a numerical simulation based on discrete Fourier analysis. Three configurations are analyzed in which the relative angles of the layers are varied. In all cases, the agreement is quite good when the distance between force and observation point is greater than a few wavelengths.     

Contour of the attenuated length of an evanescent wave at constant frequency within a band gap of photonic crystal

The plane wave method is normally applied to determine the eigenfrequency of a two-dimensional (2D) photonic crystal. A slight change to this eigenvalue equation makes the wave number its eigenvalue providing a direct means to determine the attenuated length of the evanescent modes at the frequency within the photonic band gap. The contour of the length of attenuation of the evanescent modes in a square lattice can be determined using the proposed wave number eigenvalue equation. The wave number eigenvalue equation for the two-dimensional (3D) photonic crystal can also be obtained using a derivation similar to that for the 2D photonic crystal. Possible applications of the proposed calculation-method are presented.     

Fourier transform of single eigenvalue probability density function using ensemble-averaged traces of the Hamiltonian

A determinantal identity is used to calculate the ensemble-averaged traces of the Hamiltonian. Using these averages a general expression is obtained for the Fourier transform of the single eigenvalue probability density function for all the three Gaussian ensembles for the two-dimensional case. It is shown how one can use the familiar step-up operators for the representation of a determinant. The ensemble-averaged traces are also used to derive the Fourier transform of the non-zero mean ensemble.     

Phase Recovery from the Raman Excitation Profile,Time Domain Information and Transform Theory

The phase recovery from the vibrational Raman excitation profile (REP), which contains only the modulus of the Raman amplitude, is discussed for the general situation where the Raman amplitude, with excitation energy extended in the complex plane, may have zeros in the right-half plane. The focus is on the dispersion method, with all results derived by contour integration. The new results for phase recovery, however, apply to both the dispersion and maximum entropy methods. An iterative procedure, with rapid convergence, is presented to overcome the experimental REP data being given in a limited energy range. The forward transform from the electronic absorption spectrum (ABS) to the REP and the inverse transform from the REP to the ABS are presented in a unified manner. The ubiquitous Hilbert transform is shown to be readily evaluated by the fast Fourier transform algorithm. Calculations are presented for β-carotene, a two-mode harmonic model with diffuse vibrational structure, azulene and iodobenzene to illustrate the theory. © 1997 John Wiley & Sons, Ltd.     

Dispersive elastodynamics of 1D banded materials and structures: analysis

The elastodynamics of 1D periodic materials and finite structures comprising these materials are studied with particular emphasis on correlating their frequency-dependent characteristics and on elucidating their pass-band and stop-band behaviors. Dispersion relations are derived for periodic materials and are employed in a novel manner for computing both pass-band and stop-band complex mode shapes. Through simulations of harmonically induced wave motion within a finite number of unit cells, conformity of the frequency band structure between infinite and finite periodic systems is shown. In particular, only one or two unit cells of a periodic material could be sufficient for “frequency bandedness” to carry over from the infinite periodic case, and only three to four unit cells are necessary for the decay in normalized transmission within a stop band to practically saturate with an increase in the number of cells. Dominant speeds in the scattered wave field within the same finite set of unit cells are observed to match those of phase and group velocities of the infinite periodic material within the most active pass band. Dynamic response due to impulse excitation also is shown to capture the infinite periodic material dynamical characteristics. Finally, steady-state vibration analyses are conducted on a finite fully periodic structure revealing a conformity in the natural frequency spread to the frequency band layout of the infinite periodic material. The steady-state forced response is observed to exhibit mode localization patterns that resemble those of the infinite periodic medium, and it is shown that the maximum localized response under stop-band conditions could be significantly less than in an equivalent homogenous structure and the converse is true for pass-band conditions.     

Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard

We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.     

椭圆响应强非局域非线性介质中的二维异步分数傅里叶变换及光束传输特性

研究了椭圆响应强非局域非线性介质中的光束传输问题. 结果表明:任意光束在这类介质中传输时均遵守二维异步分数傅里叶变换的传输规律. 基于二维异步分数傅里叶变换这一数学工具, 可很方便地对光束的传输进行解析求解并分析其性质. 利用二维异步分数傅里叶变换的性质, 讨论了一般光束的传输性质; 分析了孤子和二维异步呼吸子的形成条件; 得出了孤子/呼吸子的相互作用规律. 关键词： 椭圆响应 强非局域非线性 孤子 呼吸子     

Modelling of continuous and discontinuous floating slab tracks in a tunnel using a periodic approach

This paper presents a periodic approach to couple a track and a tunnel-soil system of different periodicity. The periodicity of the track and the tunnel-soil system is exploited using the Floquet transform to efficiently formulate the problem in the frequency-wavenumber domain as well as to limit the discretization effort to a reference cell. The track and the tunnel-soil system are modelled as two separate systems of different periodicity and are coupled in the frequency-wavenumber domain. A coupled periodic finite element-boundary element method is used to model the tunnel-soil system, while a periodic finite element model or an analytical approach is used to model the track.A general analytical formulation to compute the response of three-dimensional periodic media that are excited by moving loads is discussed. It is shown that the response due to moving loads on the track can be calculated from the transfer function of the track-tunnel-soil system and the axle loads.A methodology for computing the transfer functions of the coupled track-tunnel-soil system as well as the computation of dynamic forces accounting for the interaction between the moving vehicle and the periodic track are described. The model accounts for quasi-static forces as well as dynamic forces due to parametric excitation and unevenness excitation.The methodology has been used to assess the vibration isolation efficiency of continuous and discontinuous floating slab tracks. It is concluded that both continuous and discontinuous floating slab tracks have a similar efficiency in the frequency range well above the isolation frequency of the slabs, which is usually higher than the slab passage frequency. In case of discontinuous slab tracks, the parametric excitation is found to be important, which results in a poorer performance of the track at low frequencies.     

Vibration response of misaligned rotors

Misalignment is one of the common faults observed in rotors. Effect of misalignment on vibration response of coupled rotors is investigated in the present study. The coupled rotor system is modelled using Timoshenko beam elements with all six dof. An experimental approach is proposed for the first time for determination of magnitude and harmonic nature of the misalignment excitation. Misalignment effect at coupling location of rotor FE model is simulated using nodal force vector. The force vector is found using misalignment coupling stiffness matrix, derived from experimental data and applied misalignment between the two rotors. Steady-state vibration response is studied for sub-critical speeds. Effect of the types of misalignment (parallel and angular) on the vibration behaviour of the coupled rotor is examined. Along with lateral vibrations, axial and torsional vibrations are also investigated and nature of the vibration response is also examined. It has been found that the misalignment couples vibrations in bending, longitudinal and torsional modes. Some diagnostic features in the fast Fourier transform (FFT) of torsional and longitudinal response related to parallel and angular misalignment have been revealed. Full spectra and orbit plots are effectively used to reveal the unique nature of misalignment fault leading to reliable misalignment diagnostic information, not clearly brought out by earlier studies.     

Transient response with arbitrary initial conditions using the DFT

It is more economic to compute the response of linear systems with Fourier methods using fast Fourier transform algorithms than with step-by-step numerical integration methods. However, one drawback of Fourier methods is the difficulty in computing transient responses with arbitrary initial conditions (ICs). When the system is modeled with constant-parameter ordinary differential equations, the response can be obtained in closed form but, when using spectral and boundary element methods, this is no longer possible. In this paper, a technique consisting of taking advantage of the periodic character of the discrete Fourier transform to include an ad hoc force pulse to impose the ICs is proposed. The technique is presented in detail and used to compute the responses of single and multiple degree-of-freedom lumped parameter systems. The responses are compared with step-by-step integration solutions.     

二维无限深方势阱中粒子运动的路径积分解法

用路径积分的方法计算了二维无限深方势阱中粒子的传播子,并由传播函数推导出二维无限深方势阱中粒子的波函数和能量,进一步体现了路径积分与其他经典量子化方法的等价性,反映了路径积分应用于难以处理的量子力学问题的价值.     

The pseudogap mode as long-lived states of noncoherent pairs with large momenta

The mirror nesting of the Fermi contour of a quasi-two-dimensional electronic system and the presence of at least one negative eigenvalue of the Fourier transform of interaction energy are sufficient conditions for the formation of bound states of the relative motion of pairs with large total momenta. As distinct from pairing by attractive interactions, the wave functions of such pairs have alternating signs and lines of zeros that twice intersect the Fermi contour in the regions of definition of relative motion momenta. The total number of intersection points between the line of zeros and the Fermi contour is determined by symmetry of a linear combination of the wave functions of crystallographically equivalent pairs. Long-lived quasi-stationary states exist in the form of noncoherent pairs with different but close momenta and cause substantial suppression of the density of one-particle states (the appearance of a pseudogap) over a fairly wide energy range. The upper tempereture bound of the pseudogap is determined by the decay of pairs, and the lower bound, by phase coherence disturbance when pairs leave the condensate that is formed at some optimum pair momentum value owing to mirror nesting.     

Angular spectrum approach to electromagnetic wave propagation in inhomogeneous media

The angular spectrum representation of the electromagnetic wave field is employed to solve the wave propagation in a weakly inhomogeneous medium. Taking the two-dimensional spatial Fourier transform of the radiation field as well as of the dielectric constant, the angular amplitude is shown to satisfy an integro-differential equation. A similar equation is also applicable for the propagation of radiation in a non-linear medium. This integro-differential equation is solved for two specific cases of interest, namely that of a stratified medium and of a square-law medium.     

Vibro-acoustic response of orthogonally stiffened panels: The effects of finite dimensions

This paper investigates the effects of finite dimensions on the vibro-acoustic response of orthogonally stiffened panels. Three types of excitations are considered: acoustical excitation, point force excitation and random excitation by a turbulent boundary layer. In each case, a spatially windowed periodic model is compared with a Rayleigh-Ritz model where the modes of the un-stiffened panel are used as the basis functions. The latter model accounts for the reflected wave field generated at the boundaries by assuming that the panel is simply supported. On the contrary, the windowed periodic model only accounts for finiteness on sound radiation (the assumption of an infinite periodic structure is used to calculate the panel response). Numerical studies show that when the bending wavelength becomes comparable or smaller than the stiffener spacing, the periodic model is able to reproduce the results obtained with the Rayleigh-Ritz model. To complement the study, the developed models are compared with numerical simulations (finite element method) and with experimental results.     

Abundant solutions of Wick-type stochastic fractional 2D KdV equations

A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to Wick-type stochastic fractional 2D KdV equations in the white noise space. These solutions include exponential decay wave solutions, soliton wave solutions, and periodic wave solutions. Two examples are explicitly given to illustrate our approach.     

Vibration modelling of helical springs with non-uniform ends

Helical springs constitute an integral part of many mechanical systems. Usually, a helical spring is modelled as a massless, frequency independent stiffness element. For a typical suspension spring, these assumptions are only valid in the quasi-static case or at low frequencies. At higher frequencies, the influence of the internal resonances of the spring grows and thus a detailed model is required. In some cases, such as when the spring is uniform, analytical models can be developed. However, in typical springs, only the central turns are uniform; the ends are often not (for example, having a varying helix angle or cross-section). Thus, obtaining analytical models in this case can be very difficult if at all possible. In this paper, the modelling of such non-uniform springs are considered. The uniform (central) part of helical springs is modelled using the wave and finite element (WFE) method since a helical spring can be regarded as a curved waveguide. The WFE model is obtained by post-processing the finite element (FE) model of a single straight or curved beam element using periodic structure theory. This yields the wave characteristics which can be used to find the dynamic stiffness matrix of the central turns of the spring. As for the non-uniform ends, they are modelled using the standard finite element (FE) method. The dynamic stiffness matrices of the ends and the central turns can be assembled as in standard FE yielding a FE/WFE model whose size is much smaller than a full FE model of the spring. This can be used to predict the stiffness of the spring and the force transmissibility. Numerical examples are presented.     

Polarization changes and beam profile deformation of light during acousto-optic interaction in isotropic media

A spatial Fourier transform approach is proposed to investigate the effects of polarization changes and beam profile deformation of light during acousto-optic (AO) interaction in isotropic media. The behaviour of the total scattered optical fields inside the AO cell can be properly described by a vector wave equation of which the permittivity is perturbed by an acoustic wave propagating inside the medium. In the Bragg regime, using a spatial Fourier transform approach, two coupled differential equations can be derived from the wave equation to depict AO interaction in the spatial frequency domain. Analytic solutions, which comprise the effects of changing polarization, beam deformation and propagating diffraction, can be found from the coupled equations. Detailed numerical simulations, including Fourier transforming the incident light profile to calculate the spectra of the scattered light beams and, hence, their profiles in space using the inverse transform, are presented.     

Tunneling spectroscopy of a two-dimensionally periodic electron system

The tunneling current between an electron gas with a periodic potential in two dimensions and a plain two-dimensional electron system (2DES) has been studied. The strength of the periodic potential, the subband energy of the plain 2DES, and an applied in-plane magnetic field were varied, mapping the Fourier transform of the periodic wave function. Periodic peaks were observed and explained by translations in the reciprocal lattice. When the potential was strongly modulated to form an array of antidots, commensurability peaks were seen in lateral transport, but, as expected, not in tunneling.