Synthesis of highly convergent 2D and 3D finite elements for short acoustic wave simulation

New higher-order finite elements of enhanced convergence properties for acoustic wave simulation are presented in the paper. The element matrices are obtained by combining modal synthesis and optimization techniques in order to achieve minimum errors of higher modes of the computational domain. As a result, simulation models of propagating wave pulses require a smaller number of finite element divisions per wavelength compared to the conventional element model thus significantly reducing computational costs. Though finite element matrices are obtained in optimization, the resulting patterns of the matrices are versatile and further can be used in any wave propagation model. The mass matrices of the elements are diagonal, so explicit time integration schemes are applicable. The usage of new elements is especially efficient in situations where wavelengths of the simulated signal are much shorter than the dimensions of the computational domain. This is referred to as short wave propagation analysis. The results of wave propagation simulation for ultrasonic measurements are presented as application examples. The B-scans and computed dispersion curves are provided for visual interpretation of the results.     

阶梯压电层合梁的波动动力学特性

采用行波理论系统地研究了压电阶梯梁的自由振动分析以及强迫响应的分析方法. 基于分布 参数理论研究了压电阶梯梁的波传播特性，忽略柔性梁横向剪切和转动惯量的影响，给出了 梁的轴向和横向的简谐波解. 将压电阶梯梁离散化为单元，考虑压电片的刚度和质量的影响， 建立了节点散射模型. 应用位移连续和力平衡条件，推导了节点的波反射和波传递矩阵，在 此基础上，引入波循环矩阵的概念，给出波循环矩阵、波传递系数矩阵的确定方法. 应用波 循环矩阵可以有效地计算结构的固有频率. 另外，应用波传递系数研究了压电陶瓷作动器位 置对其驱动能力的影响. 得出两个主要结论：1)作动器靠近悬臂梁固定端将有较强的驱动 能力，悬臂梁边界反射行波产生弯曲消失波有利于增大压电波的模态传递系数；2)模态传 递系数与固有频率的灵敏度密切相关，波传递系数越大, 对应该处固有频率变化灵敏度越大. 另外，数值算例表明了行波方法比有限元方法具有更高的计算精度.     

Vibration analysis of a multi-span rotating ring with ray tracing method

A ray tracing method is applied to analyze both the free and forced responses in a rotating multi-span section ring. In this paper, element coordinates are established and coupled by dispersion and transmission matrices. Structure vibration displacements are expressed in a wave form with a combination of propagation, fast-attenuating and near-field waves. Meanwhile, an exciting force is considered as a point discontinues with different elements on both sides. The wave reflection and transmission matrices are introduced through coupling different elements by applying wave transmission coefficients and transfer matrices. For numeric computation, the reflection and transmission matrices are assembled, independent waveguide elements are integrated and the responses of rotating rings with non-uniform section area are derived. The structure modeling and a numeric computation with corresponding solutions illustrate the validity of the presented approach. The investigation result also shows that the presented approach can be extended to compute accurately on the dynamic characteristics of a rotating complex structure (high speed bearing cage).     

高速列车穿越有竖井隧道流场的特性研究

给出了高速列车穿越隧道压力波的三维粘性流场数值模拟过程,控制方程为三维粘性、可压缩、等熵和非定常流的Navier-Stokes方程,空间离散采用了中心有限体积法格式,时间采用预处理二阶精度多步后差分格式进行离散,对隧道壁面采用壁面函数处理。在模拟中考虑了竖井的位置、竖井的断面积、竖井的数目等因素对隧道内压力及压力梯度的影响。计算结果表明,竖井的存在改变了压缩波的波前形状,从而使得隧道内的压力变化的最大值降低,因而能够降低隧道内的压力梯度最大值,但它并不能延长压力上升的时间。     

First-order hyperbolic form of velocity-stress equations for waves in elastic solids with hexagonal symmetry

This paper reports a theoretical framework to analyze wave propagation in elastic solids of hexagonal symmetry. The governing equations include the equations of motions and partial differentiation of elastic constitutive relations with respect to time. The result is a set of nine, first-order, fully-coupled, hyperbolic partial differential equations with velocities and stress components as the unknowns. The equation set is then cast into a vector form with three 9 × 9 coefficient (or Jacobian) matrices. Physics of wave propagation are fully described by the eigen structure of these matrices. In particular, the eigenvalues of the Jacobian matrices are the wave speeds and a part of the left eigenvectors represents the wave polarization. Without invoking the plane wave solution and the Christoffel equation, two- and three-dimensional slowness profiles can be calculated. As an example, slowness profiles of a cadmium sulfide crystal are presented.     

Disperson of bending waves in rods with periodic parameters

An analytical-numerical method is developed for analysis of the geometric dispersion of bending waves in a rod with bending rigidity and mass per unit length varying periodically along its axis. The Bernoulli-Euler equations in the case of harmonic vibrations are reduced to a Hamiltonian system of the longitudinal coordinate. The general solution is constructed. It is expressed in terms of the matriciant of the system over one period, multipliers, and the eigenvectors of monodromy matrices. A technique is developed to determine the wave propagation constant as a function of the frequency, and the conditions of wave blanking and transmission are established. The results of solution and analysis of specific problems are presented. National University of Building and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 94–99, December, 1999.     

Dynamic Analysis of Large Space Structures Using Transfer Matrices and Joint Coupling Matrices

Abstract

Linear dynamic analysis of lattice structures using transfer matrices and joint coupling matrices is presented. A lattice structure is defined as a network of one-dimensional members that are connected by joints. Two examples are considered to illustrate how transfer matrices and joint coupling matrices may be used to compute natural frequencies of vibration. These two examples indicate that the transfer matrix and joint coupling matrix analysis is numerically accurate over a wide range of frequencies and becomes increasingly efficient, compared to the finite element method, as the frequency increases. Some suggestions for further improvements in computational efficiency and some comments about applicability to numerical analysis of wave propagation problems are given.     

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh-Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.     

Active Stabilization of a Rotating Shaft Transmitting Static Torque

The paper is concerned with the problem of stability of a power transmitting thin-walled shaft made of the active laminate PFC (piezoelectric fiber composite). The shaft rotates with a given operational angular velocity, and is loaded by a static torque. Such a system is known to exhibit divergence or oscillating type of instability. On the one hand presence of internal friction in the material of the shaft leads to loss of stability at a certain critical rotation speed. The static torque can be responsible for spatial deformation of the shaft axis on the other. A method preventing the system from such behavior is discussed in the paper. The method is based on application of a composite material, which contains active piezoelectric fibers. The fibers produce bending moments, and this way affect the dynamics of the entire system. Two control strategies are investigated. Results of numerical simulations are presented graphically.     

A simple spinning laminated composite shaft model

A simple spinning composite shaft model is presented in this paper. The composite shaft contains discrete isotropic rigid disks and is supported by bearings that are modeled as springs and viscous dampers. Based on a first-order shear deformable beam theory, the strain energy of the shaft are found by adopting the three-dimensional constitutive relations of material with the help of the coordinates transformation, while the kinetic energy of the shaft system is obtained via utilizing the moving rotating coordinate systems adhered to the cross-sections of shaft. The extended Hamilton’s principle is employed to derive the governing equations. In the model the transverse shear deformation, rotary inertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. To verify the present model, the critical speeds of composite shaft systems are compared with those available in the literature. A numerical example is also given to illustrate the frequencies, mode shapes, and transient response of a particular composite shaft system.     

Dynamics and stability of elastic shaft-bearing systems with nonlinear bearing parameters

The dynamics and stability of a continuously elastic spinning shaft mounted on two dissimilar end bearings possessing nonlinear anisotropic and cross coupling stiffness and damping coefficients are analyzed. Sufficient conditions of system stability in the sense of Liapunov are derived. The developed stability criteria of the considered system have been shown to reduce to that of simpler models found in the pertinent literature. The effects of nonlinearity of bearing stiffness and damping parameters together with shaft stiffness parameter and other system parameters on the dynamic stability of the system are investigated. Several graphs demonstrating parametrically the influence of various system nondimensionalized parameters on system stability boundaries for typical cases are presented. Received on 26 October 1998     

A finite wavelet domain method for wave propagation analysis in thick laminated composite and sandwich plates

An efficient numerical method is developed for the simulation of three dimensional transient dynamic response in thick laminated composite and sandwich plate structures involving very high frequencies and wave numbers. The proposed method incorporates Daubechies wavelet scaling functions for the interpolation of the in-plane displacements with a Galerkin formulation. It further explores the orthonormality and compact support of wavelet scaling functions to produce near diagonal consistent mass matrices and banded stiffness matrices. Hence, an uncoupled equivalent discrete spatial dynamic system is formulated, synthesized and rapidly solved in the wavelet domain using an explicit time integration scheme. The in-plane wavelet interpolation is further combined with an efficient high order layerwise laminate plate theory, that implements Hermite cubic splines for the through-the-thickness approximation of displacement fields. Numerical results are presented on the prediction of guided waves in laminated and thick sandwich composite plates and compared with respective solutions obtained by analytical, semi-analytical and time domain spectral element models. The method yielded higher convergence rates and substantial reductions in computational effort compared to respective time domain spectral finite elements.     

Recursive geometric integrators for wave propagation in a functionally graded multilayered elastic medium

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.     

Wideband fast multipole boundary element method: Application to acoustic scattering from aerodynamic bodies

A wideband adaptive multi‐level fast multipole method (MLFMM) is used to accelerate the matrix–vector products arising from a boundary element method (BEM) formulation which solves the Burton–Miller boundary integral equation (BIE). The wideband MLFMM presented here applies a plane wave expansion formulation with fast interpolation and filtering for calculations in the high‐frequency regime and a partial wave expansion formulation with rotation‐coaxial translation in the low‐frequency regime. The iterative solvers GMRES, Bi‐CGSTAB and CGS are tested and compared and a block diagonal preconditioner is used to improve the condition number of the BEM matrices and to accelerate the convergence of the iterative solvers. Details on the implementation of the formulations are described, including the treatment of singular integrals. Results for acoustic scattering from a wing plus engine nacelle configuration for a prescribed source in a subsonic uniform flow are presented for a broad range of frequencies in order to assess the implemented capability. Copyright © 2010 John Wiley & Sons, Ltd.     

Wave localization in randomly disordered periodic layered piezoelectric structures

Considering the mechnoelectrical coupling, the localization of SH-waves in disordered periodic layered piezoelectric structures is studied. The waves propagating in directions normal and tangential to the layers are considered. The transfer matrices between two consecutive unit cells are obtained according to the continuity conditions. The expressions of localization factor and localization length in the disordered periodic structures are presented. For the disordered periodic piezoelectric structures, the numerical results of localization factor and localization length are presented and discussed. It can be seen from the results that the frequency passbands and stopbands appear for the ordered periodic structures and the wave localization phenomenon occurs in the disordered periodic ones, and the larger the coefficient of variation is, the greater the degree of wave localization is. The widths of stopbands in the ordered periodic structures are very narrow when the properties of the consecutive piezoelectric materials are similar and the intervals of stopbands become broader when a certain material parameter has large changes. For the wave propagating in the direction normal to the layers the localization length has less dependence on the frequency, but for the wave propagating in the direction tangential to the layers the localization length is strongly dependent on the frequency.The project supported by National Natural Science Foundation of China (10632020, 10672017 and 20451057).     

In-plane wave motion and resonance phenomena in periodically layered composites with a crack

This paper investigates the transmission and propagation of two-dimensional (2D) time-harmonic plane waves in periodically multilayered elastic composites with a strip-like crack. The total wave field in the composite structure is represented as a sum of the incident wave field determined by the transfer matrix method and the scattered wave field described by integral representations in terms of the Green’s matrices and the crack-opening-displacements. A numerical scheme is developed to compute the wave propagation characteristics and the crack-characterizing quantities. The effects of the crack location and size as well as the angle of wave incidence are investigated using the averaged crack-opening-displacements and the stress intensity factors. Special attention of the paper is devoted to resonance wave motion and wave localization phenomena in a stack of periodical elastic layers weakened by a single strip-like crack. Numerical results are presented and discussed to reveal the usual and the resonant wave transmission by using the power-density vector and the energy streamlines in the vicinity of the crack. Wave localization due to interior and interface cracks is analyzed by considering the energy captured by a crack, and resonance induced crack growth is also discussed.     

Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports

Cracks appearing in the shaft of a rotary system are one of the main causes of accidents for large rotary machine systems. This research focuses on investigating the bifurcation and chaotic behavior of a rotating system with considerations of various crack depth and rotating speed of the system’s shaft. An equivalent linear-spring model is utilized to describe the cracks on the shaft. The breathing of the cracks due to the rotation of the shaft is represented with a series truncated time-varying cosine series. The geometric nonlinearity of the shaft, the masses of the shaft and a disc mounted on the shaft, and the viscoelasticity of the supports are taken into account in modeling the nonlinear dynamic rotor system. Numerical simulations are performed to study the bifurcation and chaos of the system. Effects of the shaft’s rotational speed, various crack depths and viscosity coefficients on the nonlinear dynamic properties of the system are investigated in detail. The system shows the existence of rich bifurcation and chaos characteristics with various system parameters. The results of this research may provide guidance for rotary machine design, machining on rotary machines, and monitoring or diagnosing of rotor system cracks.     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

可倾瓦径向滑动轴承完整动特性系数的分析模型

基于可倾瓦径向滑动轴承瓦块的扰动特性,提出了计算轴承完整动力系数的数学解析模型。在由单块瓦和轴颈构成的子系统上建立局部动坐标参考系,全局广义位移向量可以通过简练的步骤转换为局部动坐标系下轴颈的位移向量,利用求解固定瓦轴承动力特性的方法求得的局部动坐标系下的油膜力又可以精确地转换为全局坐标系下的表达形式,全局坐标系下的油膜力向量关于广义位移和广义速度的Jocabian矩阵形成轴承的完整动力特性系数;利用Newton-Raphson方法同时求解瓦块和轴颈在给定的静态载荷下的平衡位置,其中所需用到的系数矩阵恰好为轴承油膜力关于广义位移的Jocabian矩阵的负值,在得到平衡位置的同时可以获得轴承完整的刚度和阻尼矩阵。数值算例证明了此方法的有效性。