GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF SIMPLY-SUPPORTED TRAPEZOIDAL SHALLOW SPHERICAL SHELL ON WINKLER FOUNDATION

The idea of Green quasifunction method is clarified in detail by considering a free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem.This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation, the irregularity of the kernel of integral equations is avoided.Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

简支梯形底扁球壳弯曲问题的准格林函数方法

以简支梯形底扁球壳的弯曲问题为例,详细阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将简支扁球壳弯曲问题的控制微分方程化为两个互相耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程...     

GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF CLAMPED THIN PLATES

The Green quasifunction method is employed to solve the free vibration problem of clamped thin plates.A Green quasifunction is established by using the fundamental solution and boundary equation of the...     

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green’s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi-Green’s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob-lem. The mode shape differential equations of the free vibration problem of a simply-supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa-tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green’s function method.     

Winkler地基上薄板问题的准格林函数方法

将准格林函数方法应用到Ｗｉｎｋｌｅｒ地基上的薄板理论中,得到了一个第二类Ｆｒｅｄｈｏｌｍ积分方程。通过边界方程的适当选择,积分方程 奇异性被克服了。算例表明,本文采用的方法具有较高的精度。     

Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation

The quasi-Green’s function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green’s function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of the problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green’s formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method.     

域内具有支承及集中质量的薄板自由振动的边界元分析

本文基于薄板小挠度弯曲问题的基本解,建立了任意边界条件、域内具有支承及附有征意个集中质量的薄板自山振动的边界积分方程,文中计算了若干算例,其精度是实际工程中所允许的。     

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.     

NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE$^{\bf 1)}$

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.     

薄板稳定问题的边界元法

本文基于小挠度薄板弯曲问题的基本解,建立了求解薄板稳定问题的边界积分方程,并计算了若干算例,结果表明用边界元法求解薄板的稳定问题是行之有效的.     

饱和多孔地基与矩形板动力相互作用的非轴对称混合边值问题

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.      

A 2.5-D dynamic model for a saturated porous medium. Part II: Boundary element method

The 3-D boundary integral equation is derived in terms of the reciprocal work theorem and used along with the 2.5-D Green’s function developed in Part I [Lu, J.F., Jeng, D.S., Williams, S., submitted for publication. A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function. Int. J. Solids Struct.] to develop the 2.5-D boundary integral equation for a saturated porous medium. The 2.5-D boundary integral equations for the wave scattering problem and the moving load problem are established. The Cauchy type singularity of the 2.5-D boundary integral equation is eliminated through introduction of an auxiliary problem and the treatment of the weakly singular kernel is also addressed. Discretisation of the 2.5-D boundary integral equation is achieved using boundary iso-parametric elements. The discrete wavenumber domain solution is obtained via the 2.5-D boundary element method, and the space domain solution is recovered using the inverse Fourier transform. To validate the new methodology, numerical results of this paper are compared with those obtained using an analytical approach; also, some numerical results and corresponding analysis are presented.     

集中载荷作用下不同边界条件的弯曲厚矩形板的受迫振动

运用边界积分法研究了四边简支、两对边固定另两对边简支、四边固定三种复杂边界条件下厚矩形板的受迫振动问题,求解过程清晰,从而给出了受迫振动控制方程和挠曲面方程。通过在Matlab平台上进行数值计算,得出了图表形式的计算结果,并与有限元模拟值进行对照。研究表明,边界积分法用于求解厚矩形板的受迫振动问题的准确性,本文推导的控制方程和挠曲面方程的正确性,进而对工程实际中的各种相关问题具有一定的现实意义,也为求解此类问题提供了一种新途径,可以直接运用到工程实际中。     

Dual reciprocity hybrid radial boundary node method for Winkler and Pasternak foundation thin plate

An efficient dual reciprocity hybrid radial boundary node method is developed for the analysis of Winkler and Pasternak foundation thin plate, in which a hybrid displacement variational principle, radial point interpolation method (RPIM) and dual reciprocity method (DRM) are combined. Firstly, the hybrid displacement variational principle is developed, in which the domain variables are interpolated by two groups of symmetric fundamental solutions, while the boundary variables are interpolated by RPIM instead of the traditional moving least square, and the shape function obtained by RPIM satisfies the delta function property, so boundary conditions can be applied directly. Besides, DRM is exploited to evaluate the particular solutions of inhomogeneous terms, which can be used to transform the domain integrals arising from the inhomogeneous term into equivalent boundary integrals. Finally, some additional equations based on the DRM theory are proposed to overcome the problem that the boundary integral equations are not enough to solve all variables. This method has the advantages of both no element mesh of meshless method and dimensionality reduction of boundary element method. Numerical examples of Winkler and Pasternak foundation plates are given to illustrate that the present method is effective, accurate and it can be further expanded into practical engineering.     

各向异性平板开孔动应力集中问题的研究

采用各向异性平板弯曲波动理论及摄动方法，对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究，得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术，为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。     

正交各向异性平板开孔弹性波的衍射与动应力集中

采用各向异性平板弯曲波动方程及摄动方法，对正交各向异性无限平板开孔弹性波的衍射及动应力集中的问题进行了分析研究，得到了在稳态波作用下此种忆边界条件波动问题的渐近形式分析解。同时采用复变函数方法及保角映射技术，为求解正交各向异性无限平板开孔弹性波的衍射及动力集中问题提供了一种统一规范的分析方法。     

具有混合边界透水条件的多孔饱和半空间上刚性圆板的垂直振动

用积分变换和积分方程研究多孔饱和半空间上刚性圆板的垂直振动问题。首先应用逐次解耦方法求解多孔饱和固体的动力基本方程－Ｂｉｏｔ波动方程。然后考虑混合边界透水条件（半空间表面与圆板的接触面是不透水的,而其余表面是透水的）,建立子多孔饱和半空间上刚性圆板垂直振动的对偶积分方程,并化对偶积分方程为第二类Ｆｒｄｄｈｏｌｍ积分方程。     

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years. The method has yielded many new analytic solutions due to its rigorousness. In this study, the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge. The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied. The solution procedure incorporates separation of variables, symplectic eigen solution and superposition. The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems. The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use. The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.     

A hybrid energy approach for vibrational modelling of laminated trapezoidal plates with point supports

An investigation on the free flexural vibration of symmetric angle-ply thin trapezoidal plates continuous over arbitrarily distributed point supports is reported. A hybrid energy approach which combines the pb-2 Rayleigh-Ritz method with the Lagrangian multiplier method is proposed for the modelling of the aforementioned plate problem. The pb-2 Rayleigh-Ritz method uses a set of Ritz functions generated from the product of a two-dimensional polynomial and the equations of boundaries each raised to the power of 0, 1 or 2 corresponding to a free, simply-supported or clamped edge, respectively. The geometric boundary conditions associated with the point supports are satisfied through the use of Lagrangian multipliers. In this paper, some new solutions for the natural frequencies of several laminated trapezoidal plates with different stacking sequences and location of point supports are presented. The first known mode shapes by means of contour plots for such laminated plates are also included.