Calculating the forced response of two-dimensional homogeneous media using the wave and finite element method

The forced response of two-dimensional, infinite, homogenous media subjected to time harmonic loading is treated. The approach starts with the wave and the finite element (WFE) method where a small segment of a homogeneous medium is modelled using commercial or in-house finite element (FE) packages. The approach is equally applicable to periodic structures with a periodic cell being modelled. This relatively small model is then used, along with periodicity conditions, to formulate an eigenvalue problem whose solution yields the wave characteristics of the whole medium. The eigenvalue problem involves the excitation frequency and the wavenumbers (or propagation constants) in the two directions. The wave characteristics of the medium are then used to obtain the response of the medium to a convected harmonic pressure (CHP). Since the Fourier transform of a general two-dimensional excitation is a linear combination of CHPs, the response to a general excitation is a linear combination of the responses to CHPs. Thus, the response of a two-dimensional medium to a general excitation can be obtained by evaluating an inverse Fourier transform. This is a double integral, one of which is evaluated analytically using contour integration and the residue theorem. The other integral can be evaluated numerically. Hence, the approach presented herein enables the response of an infinite two-dimensional or periodic medium to an arbitrary load to be computed via (a) modelling a small segment of the medium using standard FE methods and post-processing its model to obtain the wave characteristics, (b) formulating the Fourier transform of the response to a general loading, and (c) computing the inverse of the Fourier transform semi-analytically via contour integration and the residue theorem, followed by a numerical integration to find the response at any point in the medium. Numerical examples are presented to illustrate the approach.     

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.     

Estimation of the loss factor of viscoelastic laminated panels from finite element analysis

A wave finite element (WFE) method is applied for predicting wave dispersion, wave attenuation and dissipation in viscoelastic laminated panels. The method involves postprocessing (using periodic structure theory) of element matrices of a small segment of the structure, which is modelled using a stack of three-dimensional finite elements meshed through the cross-section. Each layer can be discretised using either one solid element or more solid elements in order to more accurately represent interlaminar stress and strain. The finite element model of the segment of the structure is typically very small, resulting in very small computation cost. Formulations for the evaluation of the global loss factor using the WFE approach are given. In particular a formulation to calculate the average loss factor in the general case of an anisotropic component is proposed. Numerical examples are then shown. These concern the evaluation of the dispersion curves and of the global loss factor for damped laminated panels of different constructions.     

Study of inspection on metal sheet with subsurface defects using linear frequency modulated ultrasound excitation thermal-wave imaging (LFM-UTWI)

The linear frequency modulated ultrasound excitation thermal wave imaging (LFM-UTWI) was investigated on detection of subsurface defects of metal sheet. A numerical finite element analysis is carried out to calculate thermal wave signal dependence of time by linear frequency modulated ultrasonic wave excitation. Cross-correlation operation in time domain and frequency domain are used to extract the main peak value and the corresponding delay time, respectively. Fourier transform (FT) is applied to calculate the amplitude and phase angle of harmonic component of thermal wave. Experimental results show that various deep subsurface defects are readily detected using LFM-UTWI with once excitation, and LFM-UTWI has an advantage of better defect detectability compared to ultrasound lock-in thermography (ULIT).     

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.     

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.     

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.     

Vibroacoustic response of an eccentric hollow cylinder

The linear 3D elasticity theory in conjunction with the classical method of separation of variables and the translational addition theorem for cylindrical wave functions are employed to investigate the three-dimensional steady-state sound radiation characteristics of an arbitrarily thick eccentric hollow cylinder of infinite length, submerged in an unbounded ideal acoustic medium, and subjected to arbitrary time-harmonic on-surface mechanical drives. The spatial Fourier transform along the shell axis and Fourier series expansion in the circumferential direction are utilized to obtain a formal integral expression for the radiated pressure field in the frequency domain. The method of stationary phase is subsequently implemented to evaluate the integral for an observation point in the far field. The analytical results are illustrated with numerical examples in which air-filled water-submerged concentric and eccentric steel cylinders are driven by harmonic concentrated radial and transverse surface loads. Effects of excitation and cylinder eccentricity on the far-field radiated pressure amplitudes/directivities are discussed and contributions from pseudo-Rayleigh, whispering gallery, and axially guided waves are examined through selected spatial dispersion patterns. Limiting cases are considered and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature.     

Sound radiation from fluid loaded orthogonally stiffened plates

The radiation of sound from infinite fluid loaded plates is examined when the plates are reinforced with two sets of orthogonal line stiffeners. The stiffeners are assumed to be equally spaced and exert only forces on the plate. The response to a convected harmonic pressure is found by using Fourier transforms and is given in terms of the harmonic amplitudes of the stiffener forces. These forces satisfy an infinite set of simultaneous equations to which a numerical solution must be found. An expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined with particular reference to point force excitation.     

On accuracy of the wave finite element predictions of wavenumbers and power flow: A benchmark problem

This rapid communication is concerned with justification of the ‘rule of thumb’, which is well known to the community of users of the finite element (FE) method in dynamics, for the accuracy assessment of the wave finite element (WFE) method. An explicit formula linking the size of a window in the dispersion diagram, where the WFE method is trustworthy, with the coarseness of a FE mesh employed is derived. It is obtained by the comparison of the exact Pochhammer-Chree solution for an elastic rod having the circular cross-section with its WFE approximations. It is shown that the WFE power flow predictions are also valid within this window.     

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.     

Waves and energy in random elastic guided media through the stochastic wave finite element method

Energy propagation in random viscoelastic media is considered in this Letter. The forced response of uncertain waveguide subject to time harmonic loading is treated. This energy model is based on a spectral approach called the “Stochastic Wave Finite Element” (SWFE) method which is detailed in this Letter. Assuming that the random properties are spatially homogeneous in the media, the SWFE is a hybridization of the deterministic wave finite element and a parametric probabilistic approach. The proposed model is applicable in a wide frequency band with reduced time consumption. Numerical examples show the effectiveness of the proposed approach to predict the statistics of kinematic and quadratic variables of guided wave propagation. The results are compared to Monte Carlo simulations.     

Parametrical Instability of Surface Type X-modes Immersed into a Nonmonochromatic Pumping Electric Field

Parametrical excitation of surface type X-modes (STXM) at the second harmonic of electron cyclotron frequency by nonmonochromatic external alternating electric field is under consideration. STXM are the eigenmodes of a planar magnetoactive plasma waveguide structure consisting of a metal wall with dielectric coating and uniform plasma filling. An external steady magnetic field is applied along the plasma interface, so it is perpendicular to the group velocity of the considered extraordinarily polarized waves. Influence of the plasma waveguide parameters on the parametrical instability of the STXM is studied. External alternating electric field is assumed to consist of two fields with different amplitudes and frequencies. A theoretical investigation is carried out using kinetic equation for plasma particles under the conditions of weak plasma spatial dispersion and small amplitudes of external electric fields. The obtained results can be useful for research in branch of edge plasma physics.     

Numerical analysis of immersed finite cylindrical shells using a coupled BEM/FEM and spatial spectrum approach

This study numerically analyzes submerged cylindrical shells using a coupled boundary element method (BEM) with finite element method (FEM) in conjunction with the wave number theory, in which the spatial Fourier transform of surface velocity for cylinders is directly related to pressure in a far field. The acoustic loading is formulated using a symmetric complex matrix derived from a boundary integral equation where the symmetry is based on an acoustic reciprocal principle for surface acoustics. In this formulation the acoustic loading matrix is a large acoustic element whose degree of freedom is connected to the normal displacement of the vibrating structures. The coupled BEM/FEM equation is a banded, symmetric matrix, and thus its bandwidth can be minimized using a proper algorithm. This formulation significantly increases numerical efficiency. The computed normal velocity is thus transformed to wave number representation to examine acoustic radiation. A finite plane cylindrical shell, without attached stiffeners, and a shell with internal ring stiffeners are chosen to demonstrate the present analysis procedure. The far field pressure computed directly from the integral equation and predicted by wave number theory correlates closely with increasing vibrating frequency. Meanwhile, the influences of the internal ring structures on acoustic radiation are examined using the wave number theory, which helps in understanding how internal structures influence radiated noise.     

基础激励下分数阶线性系统的响应特性分析

本文研究了基础激励下含分数阶阻尼的线性系统的响应特性. 当基础激励为简谐激励时, 通过待定系数方法求得系统的动力传递系数; 当基础激励为非简谐周期激励时, 首先将激励展开成傅里叶级数, 然后根据线性系统的叠加原理求得激励中各阶频率成分所引起的动力传递系数, 并根据展开的傅里叶级数解决了数值运算中的不可导问题. 用数值仿真的方法对解析结果进行了验证, 两者符合良好, 证明了解析分析的正确性. 研究表明, 基础激励引起的动力传递系数依赖于分数阶阻尼阶数的值, 通过调节阻尼阶数可以控制动力传递系数的大小. 对于基础激励为非简谐的周期激励情况, 当激励频率一定时, 激励中的高阶频率成分引起的动力传递系数可能大于激励中的低阶频率成分引起的动力传递系数. 因此, 激励中的高阶频率成分所起的作用是不可忽略的.     

Wave based analysis of the Green's function for a layered cylindrical shell

Cylindrical shells composed of concentric layers may be designed to affect the way that elastic waves are generated and propagated, particularly when some layers are anisotropic. To aid the design process, the present work develops a wave based analysis of the Green's function for a layered cylindrical shell in which the response is given as a sum of waves propagating in the axial coordinate. The analysis assumes linear Hookean materials for each layer. It uses finite element discretizations in the radial coordinate and Fourier series expansions in the circumferential coordinate, leading to linear equations in the axial wavenumber domain that relate shell displacements and forces. Inversion to the axial domain is accomplished via a state-space formulation that is evaluated using residue integration. The resulting expression for the Green's function for each circumferential harmonic is a summation over the natural waves of the shell. The finite element discretization in the radial direction allows the approach to be used for arbitrarily thick shells. The approach is benchmarked to results from an isotropic shell and numerical examples are given for a shell composed of a fiber-reinforced material. The numerical examples illustrate the effect of fiber orientation on the Green's function.     

Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars

Based on the strain invariant relationship and taking the high-order elastic energy into account, a nonlinear wave equation is derived, in which the excitation, linear damping, and the other nonlinear terms are regarded as the first-order correction to the linear wave equation. To solve the equation, the biggest challenge is that the secular terms exist not only in the fundamental wave equation but also in the harmonic wave equation （unlike the Duffing oscillator, where they exist only in the fundamental wave equation）. In order to overcome this difficulty and to obtain a steady periodic solution by the perturbation technique, the following procedures are taken： （i） for the fundamental wave equation, the secular term is eliminated and therefore a frequency response equation is obtained; （ii） for the harmonics, the cumulative solutions are sought by the Lagrange variation parameter method. It is shown by the results obtained that the second- and higher-order harmonic waves exist in a vibrating bar, of which the amplitude increases linearly with the distance from the source when its length is much more than the wavelength; the shift of the resonant peak and the amplitudes of the harmonic waves depend closely on nonlinear coefficients; there are similarities to a certain extent among the amplitudes of the odd- （or even-） order harmonics, based on which the nonlinear coefficients can be determined by varying the strain and measuring the amplitudes of the harmonic waves in different locations.     

Analysis of wave propagation in cylindrical pipes with local inhomogeneities

In this paper, we present a numerical approach to study the guided elastic wave propagation in cylindrical pipes with local inhomogeneities. A hybrid wave finite element (WFE) and finite element (FE) technique is introduced to investigate the dispersion and wave scattering in pipes by taking full advantage of the existing FE codes. Dynamic reduction technique is employed to improve the computational efficiency, which is particularly suitable for the pipes with standard local features. Numerical examples indicate that the proposed technique provides an effective way to calculate the dispersion relationship and the scattered field. Both the axisymmetric and non-axisymmetric wave scattering problems are considered.     

S变换轮廓术中消除条纹非线性影响的方法

S变换是短时傅里叶变换和小波变换的延伸和推广,是一种无损可逆的非平稳信号时频分析方法.它不仅具有线性、多分辨性和逆变换唯一性等特点,而且其反变换与傅里叶变换保持着直接的联系.在S变换中,以简谐波作为基波,以可以同时进行伸缩和平移的高斯函数作为窗函数.同短时傅里叶变换相比,S变换的时频分辨率可以同时达到最佳,同小波变换相...     

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.