Stationary points iteration method for periodic solution to nonlinear system

The value method which is used to obtain the periodic solution to nonlinear system is mentioned in this article. Different point reflection is defined in the nonlinear autonomous and nonautonomous system firstly and then that linear reflection obtained from the inserting value of nonlinear reflection is asymptotic to original nonlinear reflection. The stationary points obtained by linear reflection are regarded as the asymptotic solution of the stationary points of original system. If this asymptotic solution of the stationary points is not satisfactorily accurate it can be used as the initial point of the next reflection. In addition, a corresponding method of researching the stability of periodic solution is put forward in this article.     

非均质中厚板的无网格LRPIM动力学分析

用局部加权残值法建立了非均质中厚板的局部径向点插值离散系统方程,采用无网格局部径向点插值法分析了非均质中厚板的自由振动和强迫振动问题。用径向基函数耦合多项式基函数来近似试函数,用四次样条函数做为加权残值法中的权函数。所构造的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件。该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行。在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化。计算结果表明,利用该方法计算非均质中厚板的自由振动和强迫振动问题可以得到具有较高精度的解。     

A variational solution of coupled vibration for moderately thick floating plate seated on the sea floor with variable water depth

Based on Reissner's plate theory, a localized variational principle for the coupled vibrations of moderately thick floating plates seated on the sea floor with variable water depth is presented in this paper. It will be a powerful tool to formulate various approximate methods of moderately thick floating plates. By means of this variational principle a formula for calculating the natural frequencies of moderately thick floating plates is also obtained. A numerical example shows that high accuracy and computational time saving may be achieved in calculation.     

厚圆板轴对称振动的弹性力学解

本文以轴对称三维弹性力学基本方程为基础,导出厚圆板强迫振动的状态方程式。利用Ｍａｃｌａｕｒｉｎ级数和Ｓｙｌｖｅｓｔｅｒ定理,厚圆板的位移和应力可以用中面位移和应力的微分算子表示。通过载荷分解和圆板表面条件,可以得到厚圆板在对称载荷与反对称载荷作用下的振动控制方程。求解了厚圆板在周边固支和简支条件下的对称与反对称的自由振动问题。通过数值计算得到了这两类自由振动的固有频率。本文的方法适用于求解厚圆板在     

Reciprocal theorem method for the forced vibration of elastic thick rectangular plates

I.IntroductionTheproblelllsofthefol.cedvibrationofelasticthickrectallgularplatestobesolvedareofgreattheoreticalsignificanceandpractical\7altle.Asthevibratinggoverningequationofthethickplateismorecomplicatedthallthatofthinplate.itisdil'ficulttoso]\'einmathematics.Manyscholarshat,eresearchedintotheseproblemsalldadvancedvariousspecialapproximations.Suchassuperpositionmethod.initialfLinctiollnletllod.thecombinedseriesot'theeigenftlnctionsofdeepbeamabouttilecorrespondillgboulldarycollditions.eller…     

An equation of motion for a thick viscoelastic plate

In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of equations of motion for the corresponding thick elastic plate, and it can be degenerated into several types of equations for various special cases.     

The vibration and stability analysis of moderate thick plates by the method of lines

The method of lines based on Hu Hai-chang's theory for the vibration and stability of moderate thick plates is developed. The standard nonlinear ordinary differential equation (ODE) system for natural frequencies and critical load is given by use of ODE techniques, and then any indicated eigenvalue could be obtained directly from ODE solver by employing the so-called initial eigenfunction technique instead of the mode orthogonality condition.Numerical examples show that the present method is very effective and reliable.     

Series solutions for a transversely loaded and completely clamped thick rectangular plate based on the three-dimensional theory of elasticity

Summary The bending of a thick rectangular plate with all the edges completely clamped is analyzed by the three-dimensional theory of elasticity. A partially distributed uniform load over the top face is dealt within the analysis. The boundary conditions of the completely clamped edges are prescribed by three-dimensional, exact conditions that three displacement components vanish at the faces. The stress distributions at the edges are minutely examined by making the best use of advantages of the exact analysis. The stress distributions for the case of a fully distributed uniform load are compared with those obtained by Reissner's theory. The values of deflections and internal forces for various thickness-length ratios are also presented. Accepted for publication 5 June 1997     

带裂纹矩形板自由振动解析解

选取带有补充项的双重正弦傅里叶级数作为振型函数通解,来解析研究带裂纹矩形板的自由振动特性。先将带裂纹矩形板分割成若干小矩形板,利用各小矩形板的边界条件,并结合振型函数中待定常数的物理意义,简化得到各小矩形板的振型函数,再结合各板的控制方程、未使用的边界条件、公共边协调条件及本文提出公共自由角点的协调条件,建立求解频率的代数方程组,然后将其转化为广义特征值问题来求解带裂纹矩形板的无量纲频率;最后选取具体参数进行计算并与文献结果对比,吻合良好,证明了本文采用的研究方法以及所提出公共角点协调条件的正确性和合理性。由于该振型函数能满足矩形板的任意边界约束,且其中的待定常数具有明确的物理意义,所以可使矩形板问题的求解统一化、简单化和规律化。     

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method （SSDQM）. The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model （ALM） is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.     

求解不连续中厚板自由振动的微分容积单元法

基于区域叠加原理和微分容积法,发展了一种新型的数值方法——微分容积单元法,用以分析具有不连续几何特征的中厚板的自由振动。根据板的不连续情况将其划分为若干单元,在每个单元内用微分容积法将控制微分方程离散成为一组线性代数方程．在相邻的单元连接处应用位移连续条件和平衡条件,引入边界约束条件后得到一套关于各配点位移的齐次线性代数方程,由此可导出求解系统固有频率的特征方程。本文用子空间迭代法求解特征方程,并以开孔板、混合边界条件板和突变厚度板为例研究了方法的收敛性和计算精度。     

Thermoelastic free vibration of clampe circular plate

An analysis was given for the free vibration of clamped circular plate when temperature and stress fields were coupled. A nonlinear differential equation about time was obtained by using Galerkin’s method. The numerical results of vibration amplitude vs time were compared with the uncoupled case. It is found that if the given initial displacement is small, the effect of thermoelastical coupling will make the natural frequency increase; if the given initial displacement is large, it will be the opposite case. Effects of some different vibration factors are also discussed.     

A solving method for a system of partial differential equations with an application to the bending problem of a thick plate

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弹性半空间地基上预应力中厚矩形板的横向振动

基于Reissner-Mindlin一阶剪切变形理论,讨论在预加面内机械荷载作用下,弹性半空间地基上四边自由中厚矩形板的横向振动问题。把地基看作三维弹性半空间体,考虑地基变形的衰减。用一组数学上完备的二元多项式作为位形函数,采用pb-2 Rayleigh-Ritz法求得四边自由中厚矩形板的自振频率和在横向简谐荷载作用下的动力响应。讨论了板的长宽比、宽厚比及弹性地基和板的相对刚度对板的自振频率的影响。     

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。     

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates.Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton's principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-ordex theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.     

Exact solutions for free vibration of transversely isotropic piezoelectric circular plates

By employing the general solution for coupled three-dimensional equations of a transversely isotropic piezoelectric body, this paper investigates the free vibration of a circular plate made of piezoelectric material. Three-dimensional exact solutions are then obtained under two specified boundary conditions, which can be used for both axisymmetric and non-axisymmetric cases. Numerical results are finally presented. The project supported by the National Natural Science Foundation of China (No. 19872060)     

Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate

The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé’s coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.     

Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation

The method of double Fourier transform was employed in the analysis of the semi-infinite elastic foundation with vertical load.And an integral representations for the displacements of the semi-infinite elastic foundation was presented.The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semi- infinite elastic foundation.Some computational results and the analysis on the influence of parameters were presented.     

基于无网格法的弹性地基加肋斜板频率数值解

为了给实际地基工程中的经济效益提供技术指标参考,应用无网格法计算加肋斜板在地基上的自由振动行为,应用移动最小二乘近似MLS和一阶剪切变形原理描述加肋斜板的位移场,分别建立斜板与肋条的势能泛函,使用Winkler弹性地基模型处理加肋板与地基之间的接触势能。通过斜板与肋条的位移协调关系寻找斜板和肋条节点参数转换公式,确立加肋斜板的自由振动控制方程。本文运用了无网格法的优势,即使肋条位置改变也不需重置网格。与有限元解的对比验证了本文方法的有效性和精确性。