A unified solution for vibration analysis of moderately thick,functionally graded rectangular plates with general boundary restraints and internal line supports

The objective of this article is to present a Fourier-Ritz method-based solution approach for the free vibration analysis of moderately thick, functionally graded (FG) rectangular plates with general boundary restraints and internal line supports. Under the current framework, regardless of boundary conditions, each of the displacements and rotations of the FG plates is invariantly expressed as a modified Fourier series in both directions. Then, the accurate solutions are obtained using the Ritz procedure based on the energy function of the plates. The convergence and accuracy of the present method are verified by a considerable number of convergence tests and comparisons.     

Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm

This paper presents a novel computational approach, the discrete singular convolution (DSC) algorithm, for analysing plate structures. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied. Approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that different methods of implementation for the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. The use of the algorithm for the vibration analysis of plates with internal supports is discussed. Detailed formulation is given to the treatment of different plate boundary conditions, including simply supported, elastically supported and clamped edges. This work paves the way for applying the DSC approach in the following paper to plates with complex support conditions, which have not been fully addressed in the literature yet. Copyright © 2002 John Wiley & Sons, Ltd.     

Free vibration analysis of thin isotropic and anisotropic rectangular plates by the discrete singular convolution algorithm

This paper presents the free vibration analysis of thin isotropic and anisotropic rectangular plates with various boundary conditions by using the discrete singular convolution (DSC) algorithm. Based on Taylor's series expansion, a unique scheme is proposed to handle various boundary conditions, including the simply supported (S), clamped (C) and free (F) edge. To validate the proposed method, the non‐dimensional frequency parameters of isotropic, orthotropic and angle‐ply symmetric laminated rectangular plates with various combinations of boundary conditions are obtained by using the DSC algorithm and compared with the analytical and/or numerical solutions. Comparisons reveal that the proposed method can handle the zero bending moment and zero shear force conditions for the isotropic as well as anisotropic plates. The proposed method provides an alternative way for applying the simply supported boundary conditions in applications of the DSC algorithm to plate structures. This investigation extends the application range of the DSC algorithm to vibration analysis of anisotropic plates with various combinations of boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Free vibration of generally supported rectangular Kirchhoff plates: State‐space‐based differential quadrature method

The title problem is investigated using the differential quadrature method based on the state‐space formalism. The plates, with mixed boundary conditions, may cross over one‐way internal rigid line supports that impose zero transverse displacement constraints. Differential quadrature procedure is applied in the direction of line supports, while exact solution is sought in the transfer domain perpendicular to the line supports using the state space method. To avoid numerical instability in the transfer matrix method, joint coupling matrices are introduced, mainly according to the continuity conditions at line joints. Natural frequencies of rectangular Kirchhoff plates with general boundary conditions are calculated and compared with the results from other methods. Effects of location of internal line supports and the mixed boundary conditions on the frequency parameters are discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

Exact analytical solutions for bending of rectangular plates with a partial internal line support

This paper studies the behavior of uniformly loaded rectangular thin plates with a partial internal line support. The highlight of the problem is that the analytical formulation explicitly considers the moment singularities that occur at the tips of partial internal line supports. The proper finite Hankel transform is used to transform a pair of dual-series equations obtained from the mixed conditions along the partial internal line support to a single Fredholm integral equation. Numerical results concerning deflection, bending moment, resultant forces, and bending-stress intensity factors are given for a square plate. Some results are also compared with the case of a square plate without partial internal line support.     

DSC‐Ritz method for the free vibration analysis of Mindlin plates

This paper introduces a novel method for the free vibration analysis of Mindlin plates. The proposed method takes the advantage of both the local bases of the discrete singular convolution (DSC) algorithm and the pb‐2 Ritz boundary functions to arrive at a new approach, called DSC‐Ritz method. Two basis functions are constructed by using DSC delta sequence kernels of the positive type. The energy functional of the Mindlin plate is represented by the newly constructed basis functions and is minimized under the Ritz variational principle. Extensive numerical experiments are considered by different combinations of boundary conditions of Mindlin plates of rectangular and triangular shapes. The performance of the proposed method is carefully validated by convergence analysis. The frequency parameters agree very well with those in the literature. Numerical experiments indicate that the proposed DSC‐Ritz method is a very promising new method for vibration analysis of Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

Vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation using element free Galerkin method

In this paper shear deformable plate theory in combination with Element-Free Galerkin Method (EFGM) is used for free vibration analysis of nonhomogeneous moderately thick plates with point supports resting on a two-parameter elastic foundation. It is shown that the vibration results obtained by this method are in a very good agreement with the available literatures in spite of using low numbers of nodes which can be considered as an inconvenience in some other methods to reach a satisfactory accuracy. Also, applicability of the method is demonstrated by solving numerical examples for different values of homogeneity variation parameter, aspect ratio, thickness to length ratio, foundation parameters, various types of boundary conditions and different numbers of point support. The numerical results present valuable information for engineers and designers in various structural applications and also prove useful to use as benchmarks for further references.     

具有中间支承的矩形板自由振动分析

应用一般解析解来求解具有中间支承矩形板的自由振动问题。一般解析解能求解任意边界条件矩形板的振动问题,求解过程是将整块板看成是沿中间支承分开的两块板,沿支承边两块板的挠度均等于零,斜度和弯矩均相等,再由全部边界条件和连续性条件可以求解各阶频率和振型。对几种具有简支边,平夹边或自由边的混合边界矩形板进行了计算。     

A unified formulation for free transverse vibration analysis of orthotropic plates of revolution with general boundary conditions

In this investigation, a general formulation for free transverse vibration analysis of orthotropic plates of revolution subjected to arbitrary boundary supports is presented by using the so-called spectro-geometric method (SGM) and the Rayleigh–Ritz technique. Under the current solution framework, the geometry of a structure can be accurately described in terms of mathematical or design parameters, rather than a computational grid or mesh. The unknown expansion coefficients are treated as the generalized coordinates and are determined using the Rayleigh–Ritz method. The accuracy and versatility of the current approach is fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions.     

边界粘弹性支承对矩形板强迫振动响应的影响

提出边界粘弹性支承对矩形板强迫振动响应影响的分析方法。该方法基于阻尼复模态分析,利用阻尼复模态的正交性,导出了板的完全解耦的模态方程;通过算例研究和讨论了粘弹性边界支承对板的振动响应的影响。     

DSC‐Ritz method for high‐mode frequency analysis of thick shallow shells

This paper addresses a challenging problem in computational mechanics—the analysis of thick shallow shells vibrating at high modes. Existing methods encounter significant difficulties for such a problem due to numerical instability. A new numerical approach, DSC‐Ritz method, is developed by taking the advantages of both the discrete singular convolution (DSC) wavelet kernels of the Dirichlet type and the Ritz method for the numerical solution of thick shells with all possible combinations of commonly occurred boundary conditions. As wavelets are localized in both frequency and co‐ordinate domains, they give rise to numerical schemes with optimal accurate, stability and flexibility. Numerical examples are considered for Mindlin plates and shells with various edge supports. Benchmark solutions are obtained and analyzed in detail. Experimental results validate the convergence, stability, accuracy and reliability of the proposed approach. In particular, with a reasonable number of grid points, the new DSC‐Ritz method is capable of producing highly accurate numerical results for high‐mode vibration frequencies, which are hitherto unavailable to engineers. Moreover, the capability of predicting high modes endows us the privilege to reveal a discrepancy between natural higher‐order vibration modes of a Mindlin plate and those calculated via an analytical relationship linking Kirchhoff and Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

转动惯量和弹性支承对非保守杆稳定性的影响

研究了具有多个弹性支承的弹性简支杆在切向均布随从力作用下的动力特性和稳定性问题。对于杆内出现的弹性支承情形,采用了以分段表示的运动微分方程、弹性支承处的连续性条件和边界条件来描述。在数值求解时,以含有两个弹性支承简支杆为例,采用有限差分法,导出了差分方程的递推格式以及边界条件和连续条件的离散形式,具体分析了弹性支承的弹性系数和支承位置以及转动惯量对非保守杆的振动频率和稳定性的影响。此外,该方法还能求解复杂边界条件下具有多个弹性支承的非保守弹性杆的复特征值问题。     

Free vibration of moving laminated composite plates

Two-dimensional axially moving materials have a wide range of industrial applications such as papers, plastics and composites in producing lines, power transmission and conveyor belts, etc. In many of these instances, the moving material is not isotropic, but is a single-layer orthotropic material or consists of several orthotropic layers.

In this article, free vibration of axially moving symmetrically laminated plates subjected to in-plan forces is studied by classical plate theory. This category includes symmetric cross-ply and angle-ply laminates and anisotropic plates. Firstly, an exact method is developed to analyze vibration of multi-span traveling cross-ply laminates, and then a semi-analytical finite strip method is extended for moving symmetric laminated plates in general, with arbitrary boundary conditions. By the finite strip method intermediate elastic or rigid supports can also be added to the model of the moving plate. The supports may be in the form of point, line or local distributed supports.     

Natural vibrations of anisotropic plates with several internal line hinges

This paper deals with the free transverse vibration of anisotropic plates with several arbitrarily located internal line hinges and piecewise smooth boundaries, elastically restrained against rotation and translation. The equations of motion and the associated boundary and transition conditions are derived using Hamilton’s principle in a rigorous framework. A new analytical manipulation based on a condensed notation is used to compact the corresponding analytical expressions. A combination of the Ritz method and the Lagrange multipliers method with polynomials as coordinate functions is used to obtain tables of the nondimensional frequencies and the corresponding mode shapes, for rectangular plates with different boundary conditions and restraint conditions in the internal line hinges. The cases not previously treated of two- and three line hinges are particularly analyzed.     

DSC analysis of a simply supported anisotropic rectangular plate

The discrete singular convolution (DSC) algorithm is used to analyze the deflection and free vibration behavior of a simply supported anisotropic rectangular plate. A novel approach is proposed to solve the difficulty in using DSC to handle the simply supported boundary conditions with bending–twisting coupling. DSC results are presented for bending under distributed load and a center concentrated load, and natural frequencies of flexural vibrations. It is shown that the DSC with proposed method to apply the simply supported boundary conditions yields very accurate results as compared to exact solutions or results obtained by methods of differential quadrature and finite element with fine meshes. It is also verified that neglecting the bending–twisting coupling in applying the simply supported boundary conditions may result incorrect solutions, especially for the bending analysis of anisotropic plates.     

具有中间支承的正交异性矩形板自由振动分析

应用一般解析解来求解具有中间支承的正交异性矩形板的自由振动问题。一般解析解能求解任意边界条件矩形板的振动问题。求解过程是将整块板看成是沿中间支承分开的两块板。沿支承边两块板的挠度均等于零;斜度和弯矩均相等。由全部边界条件和连续性条件可以求解各阶频率和振型。对几种具有简支边、平夹边或自由边的板进行了计算。     

A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.     

Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first‐order shear deformation theory

In this study, free vibration of laminated skew plates was investigated. Discrete singular convolution (DSC) method is used for numerical solution of vibration problems. The straight‐sided quadrilateral domain is mapped into a square domain in the computational space using a four‐node element by using the geometric transformation. Typical results are presented for different geometric parameters and boundary conditions. It is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate. The results obtained by DSC method are compared with those obtained by analytical and numerical approaches. It is shown that reasonable accurate results are obtained. Present work also indicates that the method of DSC is a promising and potential approach for computational mechanics. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional vibration analysis of rectangular thick plates on Pasternak foundation

The free‐vibration characteristics of rectangular thick plates resting on elastic foundations have been studied, based on the three‐dimensional, linear and small strain elasticity theory. The foundation is described by the Pasternak (two‐parameter) model. The Ritz method is used to derive the eigenvalue equation of the rectangular plate by augmenting the strain energy of the plate with the potential energy of the elastic foundation. The Chebyshev polynomials multiplied by a boundary function are selected as the admissible functions of the displacement functions in each direction. The approach is suitable for rectangular plates with arbitrary boundary conditions. Convergence and comparison studies have been performed on square plates on elastic foundations with different boundary conditions. It is shown that the present method has a rapid convergent rate, stable numerical operation and very high accuracy. Parametric investigations on the dynamic behaviour of clamped square thick plates on elastic foundations have been carried out in detail, with respect to different thickness–span ratios and foundation parameters. Some results found for the first time have been given and some important conclusions have been drawn. Copyright © 2004 John Wiley & Sons, Ltd.