边界约束刚度不确定的结构振动特征值

利用摄动法 ,将随机的微分方程和边界条件化为一系列的确定性微分方程和边界条件。运用有限元离散方法 ,推导了统计特征值的二阶摄动近似表达 ,用算例对本文方法进行了说明并和 Monte-Carlo模拟法结果进行了比较     

Analytical investigation for true and spurious eigensolutions of multiply-connected membranes containing elliptical boundaries using the dual BIEM

It is well known that the boundary element method may induce spurious eigenvalues while solving eigenvalue problems. The finding that spurious eigenvalues depend on the geometry of inner boundary and the approach utilized has been revealed analytically and numerically in the literature. However, all the related efforts were focused on eigenproblems involving circular boundaries. On the other hand, the extension to elliptical boundaries seems not straightforward and lacks of attention. Accordingly, this paper performs an analytical investigation of spurious eigenvalues for a confocal elliptical membrane using boundary integral equation methods (BIEM) in conjunction with separable kernels and eigenfunction expansion. To analytically study this eigenproblem, the elliptic coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the separable kernel by using the elliptic coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the separable kernel, boundary density and boundary contour integration and they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integration. In this way, a similar finding about the mechanism of spurious eigenvalues is found and agrees with those corresponding to the annular case. To verify this finding, the boundary element method and the commercial finite-element code ABAQUS are also utilized to provide eigensolutions, respectively, for comparisons. Good agreement is observed from comparisons. Based on the adaptive observer system, the present approach can deal with eigenproblems containing circular and elliptical boundaries at the same time in a semi-analytical manner. By using the BIEM, it is found that spurious eigenvalues are the zeros of the modified Mathieu functions which depend on the inner elliptical boundary and the integral formulation. Finally, several methods including the CHIEF method, the SVD updating technique and the Burton & Miller method are employed to filter out the spurious eigenvalues, respectively. In addition, the efficiency of the CHIEF method is better than those of the SVD updating technique and the Burton & Miller approach, since not only hypersingularity is avoided but also computation effort is saved.     

Fluid sloshing in an ice-hole of arbitrary shape

A numerical method is proposed for determining the natural frequencies and modes of the small oscillations of an ideal fluid in a half-space bounded above by a rigid plane with an aperture of arbitrary shape. Considering the monotonic dependence of the eigenvalues on the geometry, it can be stated that the eigenvalues for a half-space are universal upper limits for the corresponding eigenvalues of tanks with an arbitrary boundary but the same free surface.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 108–112, July–August, 1992.     

Closed form stress distribution in 2D elasticity for all boundary conditions

This paper applies a Hamiltonian method to study analytically the stress dis- tributions of orthotropic two-dimensional elasticity in(x,z)plane for arbitrary boundary conditions without beam assumptions.It is a method of separable variables for partial differential equations using displacements and their conjugate stresses as unknowns.Since coordinates(x,z)can not be easily separated,an alternative symplectic expansion is used. Similar to the Hamiltonian formulation in classical dynamics,we treat the x coordinate as time variable so that z becomes the only independent coordinate in the Hamiltonian ma- trix differential operator.The exponential of the Hamiltonian matrix is symplectic.There are homogenous solutions with constants to be determined by the boundary conditions and particular integrals satisfying the loading conditions.The homogenous solutions consist of the eigen-solutions of the derogatory zero eigenvalues(zero eigen-solutions) and that of the well-behaved nonzero eigenvalues(nonzero eigen-solutions).The Jordan chains at zero eigenvalues give the classical Saint-Venant solutions associated with aver- aged global behaviors such as rigid-body translation,rigid-body rotation or bending.On the other hand,the nonzero eigen-solutions describe the exponentially decaying localized solutions usually ignored by Saint-Venant's principle.Completed numerical examples are newly given to compare with established results.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

The Discrete Cosserat Eigenfunctions for a Spherical Shell

The Papkovich-Neuber potential method is applied to obtain the discrete Cosserat eigenvalues and eigenvectors for the boundary value problems of displacement and traction for a spherical shell. The eigenvalues presented herein correct those obtained by the Cosserats. This revised version was published online in August 2006 with corrections to the Cover Date.     

ON THE STABILITY BOUNDARY OF HAMILTONIAN SYSTEMS

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Generation of Three-Dimensional Disturbances in a Boundary Layer on Strong Interaction with a Hypersonic Flow

Laminar boundary layer flow over an infinite-span, finite-length flat plate is investigated in the regime of strong interaction with a hypersonic gas flow. Under the assumption that an additional condition dependent on the transverse coordinate can be imposed on the trailing edge of the plate the flow functions are expanded in power series in the vicinity of the leading edge. It is shown that these expansions include an indefinite function dependent on the transverse coordinate. The corresponding boundary value problems are formulated and solved and the eigenvalues are determined. It is established that in this case the two-dimensional boundary layer can rearrange itself into a three-dimensional boundary layer.     

Effect of magnetic field and non-homogeneity on the radial vibrations in hollow rotating elastic cylinder

In this paper, the natural frequencies of the radial vibrations of a hollow cylinder with different boundary conditions under influences of magnetic field, rotation and non-homogeneity have been studied. The solution of the problem is obtained by using technique of variables separation. In the present paper three different boundary conditions are considered, namely the free, fixed and mixed boundary conditions. The displacement and stresses components have been obtained in analytical form involving Bessel function of first and second kind and of order n. The determination is concerned with the eigenvalues of the natural frequencies of the radial vibrations for different boundary conditions in the case of harmonic vibrations. Numerical results are given and illustrated graphically for each case considered. Comparisons are made with the results in the absence of magnetic field, rotation and non-homogeneity. The results indicate that the effect of magnetic field, rotation and non-homogeneity are very pronounced.     

A practical method for the evaluation of eigenfunctions from compound matrix variables in finite elastic bifurcation problems

We show how the compound matrix method can be extended to give eigenfunctions as well as generalised eigenvalues to bifurcation problems in non-linear elasticity. When the incremental problem is formulated in terms of displacements only there are significant difficulties that arise from the non-trivial boundary conditions. In order to avoid these problems we adopt a Stroh formulation of the incremental problem. This then produces trivial boundary conditions for the compound matrix eigenvalue problem and more importantly known initial conditions for the compound matrix eigenfunction problem. This results in a straightforward and robust calculation for the eigenfunctions.     

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples. The project supported by the National Science Foundations of Russia and China (10072012)     

哈密顿体系与弹性楔体问题

将哈密体系引入到级坐标下的弹性力学楔体问题,利用该体系辛空间的性质,将问题化为本征值和本征向量求解上,得到了完备的解空间,从而改变了弹性力学传统的拉格朗日体系以应力函数为特征的半逆法的讨论去解决该类问题的思路,给出了一条求解该类问题的直接法。     

Aeroelastic analysis of helicopter rotor blade in hover using an efficient reduced-order aerodynamic model

This paper presents a coupled flap–lag–torsion aeroelastic stability analysis and response of a hingeless helicopter blade in the hovering flight condition. The boundary element method based on the wake eigenvalues is used for the prediction of unsteady airloads of the rotor blade. The aeroelastic equations of motion of the rotor blade are derived by Galerkin's method. To obtain the aeroelastic stability and response, the governing nonlinear equations of motion are linearized about the nonlinear steady equilibrium positions using small perturbation theory. The equilibrium deflections are calculated through the iterative Newton–Raphson method. Numerical results comprising steady equilibrium state deflections, aeroelastic eigenvalues and time history response about these states for a two-bladed rotor are presented, and some of them are compared with those obtained from a two-dimensional quasi-steady strip aerodynamic theory. Also, the effect of the number of aerodynamic eigenmodes is investigated. The results show that the three-dimensional aerodynamic formulation has considerable impact on the determination of both the equilibrium condition and lead-lag instability.     

Time-dependent fiber spinning equations. 2. Analysis of the stability of numerical approximations

The truncated time-dependent one-dimensional fiber spinning equations for an Upper Convected Maxwell model, presented in the previous paper [1], are approximated using various numerical methods. The eigenvalues and eigenfunctions of the linearized problem around the steady-state solution are calculated and compared against the analytical results for various values of Deborah and Reynolds numbers.It is shown that upwind formulations of Finite-Difference and Finite-Element methods produce the most stable results. The most accurate eigenvalues and eigensolutions are those calculated by Pseudospectral methods. However, both Chebyshev and Legendre Pseudospectral methods become unstable at high values of De. Finally, it is shown that the boundary conditions have to be incorporated into the numerical scheme in a suitable way; spectacular instabilities might be generated otherwise.     

Perturbation analysis of a continuous system for desalination by reverse osmosis

Blasius perturbation solutions are developed for the incompressible laminar boundary layer at high Schmidt numbers. The eigenvalues have been found to be multiples of 3/2 and the eigenfunctions can be expressed as confluent hypergeometric functions. The eigenfunctions are employed in analyzing a continuous desalination system. The reverse-osmosis boundary condition leads to an integral equation which is, in turn, solved by iterative techniques. The first-order calculations show that the results are most accurate for large ratios of osmotic to applied pressures. For medium ratios, the results are reasonably good.The present analysis should be useful, in general, for high Schmidt number flows with linear or nonlinear surface conditions provided that the transverse wall velocity is vanishingly small.     

Two-dimensional piezoelectricity. Part I: eigensolutions of nondegenerate and degenerate materials

General solutions of two-dimensional piezoelectricity, which yield all solutions of 2-D boundary values problems, are obtained by combining four complex conjugate pairs of independent eigensolutions, each containing an arbitrary analytic function. The forms of representation are fundamentally different for 14 different classes of nondegenerate and degenerate piezoelectric materials, as determined by the multiplicity and types of eigenvalues. Degenerate materials possess high-order eigensolutions, in which the eigenvectors of equal and lower orders are intrinsically coupled. Such coupling is nonexistent in nondegenerate cases including the well-known and analytically simple case with no multiple eigenvalues. The present analysis is drastically simplified by using the compliance-based formalism, instead of the stiffness-based, extended Eshelby–Stroh formalism. Explicit expressions are obtained for the eigensolutions, the pseudometrics, and the intrinsic tensors characterizing piezoelectric materials of every type.     

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems. Foundation item: the National Natural Science Foundation of China (10072012); the National Natural Science Foundation of Russia Biography: QI Zhao-hui (1964-)     

广义边界梁的动力学建模与动态特征

对于广义边界条件Euler-Bernoulli梁,采用相对描述方式建立了可描述梁整体运动和相对变形的几何非线性及其线性化动力学模型,应用线性变换得到了该类梁的线性经典动力学方程,得到了广义边界条件下梁的横向振动代数特征方程、特征函数及特征值的退化表达式.算例分析了边界小扰动对固支-固支梁横向振动特征的影响规律.     

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

A multiple monopole （or multipole） method based on the generalized mul- tipole technique （GMT） is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure is obtained. Some numerical examples are presented to validate the proposed method.     

Antiplane stress singularities in a bonded bimaterial piezoelectric wedge

Summary  In this paper, the eigen-equations governing antiplane stress singularities in a bonded piezoelectric wedge are derived analytically. Boundary conditions are set as various combinations of traction-free, clamped, electrically open and electrically closed ones. Application of the Mellin transform to the stress/electric displacement function or displacement/electric potential function and particular boundary and continuity conditions yields identical eigen-equations. All of the analytical results are tabulated. It is found that the singularity orders of a bonded bimaterial piezoelectric wedge may be complex, as opposed to those of the antiplane elastic bonded wedge, which are always real. For a single piezoelectric wedge, the eigen-equations are independent of material constants, and the eigenvalues are all real, except in the case of the combination C–D. In this special case, C–D, the real part of the complex eigenvalues is not dependent on material constants, while the imaginary part is. Received 26 March 2002; accepted for publication 2 July 2002