A PARAMETRIC STUDY ON THE AXISYMMETRIC MODES OF VIBRATION OF MULTI-LAYERED SPHERICAL SHELLS WITH LIQUID CORES OF RELEVANCE TO HEAD IMPACT MODELLING

The free vibration of spheres composed of inviscid compressible liquid cores surrounded by spherical layers of linear elastic, homogeneous and isotropic materials are studied using three-dimensional elasticity equations. The exact three-dimensional equations are first derived for an N -layered sphere with a liquid core and an extensive parametric study is then presented for the first few natural frequencies of the spheroidal modes of vibration. Non-dimensional frequency parameters are compared with values obtained using lower order membrane and shell theories. It is shown that for a remarkably wide range of geometric and material parameters, which encompasses values typical for the human head, the first ovalling mode of a fluid-filled shell behaves like a membrane filled with incompressible fluid and a simple closed-form expression is derived which closely approximates the natural frequencies obtained using the exact three-dimensional equations.     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

考虑铺层角的复合材料圆柱壳自由振动三维弹性准确解

为了获得任意角度铺层的多层复合材料圆柱壳的自由振动准确解,在三维弹性理论的基础上,结合分层理论和状态空间法,建立横向位移和应力的传递矩阵,轴向和环向位移采用双螺旋模式的位移函数,对任意角度铺层复合材料圆柱壳简支边界条件下的自由振动进行了理论推导,得到了自由振动方程的精确形式。与文献理论解和有限元计算结果对比,结果表明,关注频率在2倍的环频率以下时,薄壳的固有频率计算精度能控制在1%以内,厚壳的固有频率计算精度能控制在2%以内。对于厚壳的计算可将壳体沿厚度方向划分为多层来处理,这样能有效提高计算精度。计算分析了铺层角对壳体固有频率的影响,环向模态数较低时,固有频率随着铺层角的增加呈抛物线变化趋势;环向模态数较高时,固有频率随着铺层角的增大单调递增。该理论方法同样适用于均质各向同性壳和正交各向异性圆柱壳。      

Sound scattering from underwater elastic spherical shell covered with multilayered medium approximated acoustic cloak

The anechoic performance and mechanism of underwater elastic spherical shell covered with coating are studied at low frequencies.The acoustic cloak is anisotropic material,which can be designed with homogeneous isotropic materials on the basis of effective medium approximation theory.The analytic expression of scattering acoustic field from the shell covered with multilayered medium is formulated and the scattering form function,resonance mode,acoustic field distribution are computed,the scattering characteristics and mechanism of transmission are analyzed.The results show that the direction of sound transmission inside the multilayered medium is changed,the acoustic field is deflected gradually,and the acoustic energy flux is guided around the target,which reduces the scattering intensity at low frequencies,the acoustic intensity of target＇s surface is very weak.Excepting the first resonance peak in spectrum produced by the zero order partial wave,the other resonance modes of elastic spherical shell are not excitated and the multilayered medium can suppress the resonance of the spherical shell effectively.     

Free vibrations of a laminated beam by a microstructure theory

The free vibrations of a laminated beam are considered within the framework of a theory that models the composite beam as a macrohomogeneous beam with microstructure. The beams are assumed to consist of several parallel alternating layers of two homogeneous, isotropic elastic materials. The system of three coupled partial differential equations is solved exactly, and attention is devoted to the determination of natural frequencies of vibration of laminated beams with (i) hinged-hinged ends and (ii) clamped-clamped ends. For the sake of comparison, the same boundary value problems are also solved within the framework of the so-called effective modulus theory, which treats the composite as a transversely isotropic and “fictitiously” homogeneous Timoshenko beam, with effective moduli and density. For relatively long beams, i.e., in the low frequency range, the natural frequencies obtained from the two theories are in excellent agreement, but as the depth-to-length ratio, ζ, increases the microstructure frequencies are observed to be much lower than the effective modulus frequencies, the magnitude of the effect becoming more pronounced with increasing mode number n.     

黏弹介质包裹的充液管道中低频轴对称波传播特性

研究埋地充液管道中低频轴对称波传播特性。将土壤考虑为黏弹介质,结合Kennard薄壳方程和Kelvin-Voigt线性黏弹性模型,引入土壤载荷矩阵,推导出土-管滑移情形下流体主导波和管壁压缩波的相速度表达式。通过数值模拟计算得到流体主导波和管壁压缩波的频散和衰减曲线并进行可靠性验证,分析两种波引起的管壁径向位移之比,讨论厚径比和品质因子对流体主导波传播的影响。结果表明,黏弹介质对流体主导波和管壁压缩波的相速度影响较小,但对衰减影响较大;流体主导波对管壁径向位移有较大的影响,是泄露噪声传播的主要载体;厚径比越大,流体主导波的相速度越大,衰减越小;而品质因子越大,流体主导波的频散和衰减都越小。研究结果可为埋地充液管道的泄漏检测提供一定的理论参考。      

On the flexural vibration of cylinders under axial loads: Numerical and experimental study

The flexural vibration of a homogeneous isotropic linearly elastic cylinder of any aspect ratio is analysed in this paper. Natural frequencies of a cylinder under uniformly distributed axial loads acting on its bases are calculated numerically by the Ritz method with terms of power series in the coordinate directions as approximating functions. The effect of axial loads on the flexural vibration cannot be described by applying infinitesimal strain theory, therefore, geometrically nonlinear strain–displacement relations with second-order terms are considered here. The natural frequencies of free–free, clamped–clamped, and sliding–sliding cylinders subjected to axial loads are calculated using the proposed three-dimensional Ritz approach and are compared with those obtained with the finite element method and the Bernoulli–Euler theory. Different experiments with cylinders axially compressed by a hydraulic press are carried out and the experimental results for the lowest flexural frequency are compared with the numerical results. An approach based on the Ritz formulation is proposed for the flexural vibration of a cylinder between the platens of the press with constraints varying with the intensity of the compression. The results show that for low compressions the cylinder behaves similarly to a sliding–sliding cylinder, whereas for high compressions the cylinder vibrates as a clamped–clamped one.     

Influence of initial geometrical imperfections on vibrations of axially compressed stiffened cylindrical shells

The vibrations of stiffened cylindrical shells having axisymmetric or asymmetric initial geometrical imperfections and axial preload are analyzed. The analysis is based on a solution of the von Kárman-Donnell non-linear shell equations, an “exact” solution of the compatibility equation, and a first order approximation by the Galerkin method of the equilibrium equation. The stiffeners are closely spaced and “smeared” stiffener theory is employed. The results of an extensive parametric study carried out on shells similar to those used in vibration and buckling tests at the Technion show that stiffening of the shell will lower the imperfection-sensitivity of its free vibrations, but the decrease depends on the type of stiffening (stringers or rings), the mode shapes of the vibration and the imperfection, the stiffener strength and eccentricity. The imperfection-sensitivity decrease, caused by the stiffeners, is greater for vibration mode shapes with high imperfection-sensitivity than for other vibration mode shapes. The sensitivity differences between stringer and ring-stiffened shells depend especially on the vibration and the imperfection mode shapes, and on their coupling. Small imperfections change the natural frequencies of stiffened shells in the same directions as for isotropic shells, but to a smaller extent. The frequency dependence on the external load is also strongly affected by the imperfection mode shape. The results correlate well with earlier ones for isotropic shells.     

Elastic fields of a screw superdislocation with a hollow core (pipe) perpendicular to the free crystal surface

Screw superdislocations with hollow cores (pipes) which arise during growth of semiconductor crystals such as silicon carbide and gallium nitride are considered. Exact analytical expressions are first derived for the displacements, strains, and stresses associated with a pipe oriented perpendicular to the free planar surface of an elastically isotropic half-space. It is shown that the stress field of the dislocation present in the pipe is heavily affected when the exact boundary conditions at the free cylindrical surface of the pipe are taken into account. The influence is the strongest in the region around the pipe at distances of the order of the pipe radius from the surface of the pipe. In this region, the elastic strains can be as large as a few tenths of one percent. The results obtained can be useful in analyzing the interaction of pipes with one another and with other defects, as well as in simulating the behavior of pipes during crystal growth.     

On the modal density and damping of cylindrical pipes

Measurements of the resonant frequencies and quality factors of a series of long, small diameter cylindrical pipes are presented. The dependence of the modal densities, calculated from these measurements, on pipe length, wall thickness and pipe material is in close agreement with theoretical statistical predictions. Measurements of the damping of steel pipes for several different end conditions are also presented. Different, but always well-ordered, variation of modal quality factor with mode order is found in each case. For free-free ends the modal quality factors are large (>1000) and determined by internal material damping, except for modes for which the acoustic radiation damping is large; the effect of radiation damping is most important for those modes which have high wave speeds at low frequencies. The quality factors for rigid end conditions are similar to those for free ends, except for translational modes of low axial order which are damped by vibration of the end supports. For end conditions which allow relative motion between the pipe and the end supports, there is considerable additional damping, probably ascribable to gas pumping in the joints.     

2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures

This paper focuses on the dynamic behavior of functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The first-order shear deformation theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. The homogeneous isotropic material is inferred as a special case of functionally graded materials (FGM). The governing equations of motion, expressed as functions of five kinematic parameters, are discretized by means of the generalized differential quadrature (GDQ) method. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. For the homogeneous isotropic special case, numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. Different typologies of non-uniform grid point distributions are considered. Finally, for the functionally graded material case numerical results illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behavior of shell structures.     

Natural frequencies and mode shapes of flexural vibration of plates: laser-interferometry detection and solutions by Ritz's method

This paper presents an experimental and theoretical study of flexural symmetric vibration modes of a linear elastic plate. A laser interferometer is used as detector of the free vibration of a rectangular parallelepiped-shaped aluminium plate. The vibration spectrum gives the lowest natural frequencies of the sample. Assumption that the vibration of the plates may be described by some approximate theories is proven to be inconsistent. The Ritz method, with products of powers of the co-ordinates as basis functions, is applied to obtain the lowest flexural natural frequencies. Three-dimensional solutions are obtained, unlike those provided by simpler theories. The experimental results are compared with the numerical predictions and a good agreement is obtained. Finally, forced motion is applied to the centre of the plate and the out-of-plane and in-plane displacement components for the first symmetric mode are measured. A good fit of the calculated values to the experimental values is found.     

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.     

A study of vibroacoustic coupling between a pump and attached water-filled pipes

This paper presents a model for the vibroacoustical behavior of a pump coupled with water-filled pipes. Coupling between (a) the pump and the inlet and outlet pipes, and (b) the pipe wall and the fluid contained in the pipe, is investigated through analytical modeling and numerical simulation. In the model, the pump is represented by a rigid body supported by multiple elastic mounts, and the inlet and outlet pipes by two semi-infinite water-filled pipes. The vibration characteristics of the coupled system under the excitation of mechanical forces and fluid-borne forces at the pump are calculated. The results enhance our understanding about how the input mechanical and fluid excitation energy at the pump is transmitted to the pipes and how to relate the piping vibroacoustical response to the excitations at the pump. This study assists in predicting dynamic stress in pipes for given excitations at the pump, and in developing methods to identify the nature (fluid or mechanical) of the excitation forces at the pump using the vibration and dynamic pressure measurements on the pump/pipe system.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

爆炸载荷作用下焊缝区附近埋地X70钢管的动力响应分析

爆破地震效应对埋地管线的影响已成为工程爆破领域研究的热点。基于有限元软件ANSYS/LS-DYNA,以两种含Y型焊缝(坡口有2 mm余高焊缝和坡口无余高焊缝)的埋地X70钢管为例,数值模拟研究了TNT炸药量相同(4.473 kg)而炸高(60.0、85.0和110.0 cm)不同时,焊缝区附近埋地X70钢管的动力响应规律。研究表明:当炸高为60.0 cm时,焊缝有余高的管道受应力集中的影响较大,且先于焊缝无余高的管道进入屈服阶段;当炸高为60.0和85.0 cm时,焊缝有余高的管道整体抵抗变形的能力明显弱于焊缝无余高的管道。管土间的相互作用对X70管道背爆面有支撑作用,可有效地减小管道背爆面的位移。在相同条件下,焊缝有余高的X70管道抵抗振动的性能弱于焊缝无余高的管道,且与焊缝形式相比,炸高对含焊缝区的X70管道的最大振速起主要影响作用。     

Free vibration of a conical shell with variable thickness

An analysis is presented for the free vibration of a truncated conical shell with variable thickness by use of the transfer matrix approach. The applicability of the classical thin shell theory is assumed and the governing equations of vibration of a conical shell are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined by quadrature of the equations, the natural frequencies and the mode shapes of vibration are calculated numerically in terms of the elements of the matrix under any combination of boundary conditions at the edges. The method is applied to truncated conical shells with linearly, parabolically or exponentially varying thickness, and the effects of the semi-vertex angle, truncated length and varying thickness on the vibration are studied.     

WAVENUMBER PREDICTION OF WAVES IN BURIED PIPES FOR WATER LEAK DETECTION

Water leaks are a topic of great concern in Britain and many other countries, because of decreasing water supplies and the deterioration of old pipework. Correlation techniques are widely used in leak detection, but for these to be effective, the propagation wavespeeds and wave attenuation must be known. Relatively predictable for metal pipes, these are largely unknown for the newer plastic pipes, being highly dependent on the pipe wall properties and the surrounding medium. In this paper, pipe equations for n=0 axisymmetric wave motion are derived for a fluid-filled pipe, surrounded by an infinite elastic medium which can support both longitudinal and shear waves. These equations are solved for two wave types,s =1,2, which correspond to a fluid dominated wave and an axial shell wave, and expressions for a complex wavenumber for each wave are given.     

Leak noise propagation and attenuation in submerged plastic water pipes

Detection of water leaks in buried distribution pipes using acoustic methods is common practice in many countries. Correlation techniques are widely used in leak detection, and these have been extremely effective when attempting to locate leaks in metal pipes. However, a number of difficulties have been highlighted when trying to determine the position of leaks in plastic pipes. Of particular interest here is what happens to the leak noise when the pipe passes through an expanse of water, such as across a river or lake.In this paper, the low-frequency acoustic propagation and attenuation characteristics of a submerged plastic water pipe are investigated experimentally in the laboratory, supported by predictions from a theoretical model. It is found that, whilst the signal attenuation for a submerged pipe is increased relative to that for a similar in-vacuo pipe, energy does not, in fact, radiate into the water; furthermore, the attenuation is small compared with that for a pipe buried in soil.     

近似声斗篷的多层介质覆盖水中弹性球壳的声散射

对水中弹性球壳的低频覆盖层消声效果和消声机理进行研究。声斗篷(Acoustic cloak)是各向异性的超常材料,根据有效介质近似理论(Effective MediumApproximationTheory),采用各向同性多层介质近似各向异性材料的声斗篷,推导出覆盖多层介质水中弹性球壳散射声场的解析表达式,计算了弹性球壳覆盖多层介质前后的散射形态函数、谐振模态以及声场分布,分析了覆盖前后球壳的声散射特性和声传播机理。研究表明,覆盖层内的声波在多层介质之间的声传播方向发生改变,声场形成弯曲变形,声能流绕过目标,这不仅极大的降低了低频散射强度,而且使到达弹性体表面的声场强度非常小,散射频响中除了0阶子波产生的第一个谐振峰外,无法激发弹性球壳的其它谐振模态,有效的抑制弹性球壳的谐振散射。