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.     

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.     

Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: Applications

Part 2 of this series of two papers presents the applications of the discrete singular convolution (DSC) algorithm. The main purpose of this paper is to explore the utility, test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports. Both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions. The effects of different size, shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated. The partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports. Several closed‐loop supports, such as ring, square and rhombus, and their combinations are studied for complex internal supports. Convergence and comparison studies are carried out to establish the correctness and accuracy of the DSC algorithm. The DSC results are compared with those in the available literature obtained by using other methods. Numerical results indicate that the DSC algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries, boundary conditions and support conditions. Copyright © 2002 John Wiley & Sons, Ltd.     

Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method

Free vibration characteristics of thick skew plates reinforced by functionally graded carbon nanotubes (CNTs) reinforced composite are presented. Discrete singular convolution (DSC) method is used for the numerical solution of vibration problems via geometric mapping technique. Using the geometric transformation via a four-node element, the straight-sided quadrilateral physical domain is mapped into a square domain in the computational space. Then the method of discrete singular convolution with some singular kernels such as Regularized Shannon's delta (RSD) and Lagrange's delta kernels (LDK) have been used for spatial discretizing of the resulting governing equation of motion. Calculated results have been presented in order to show the effects of volume fraction of CNT, skew angles, CNT distribution types, plate aspect ratio and length-to-thickness ratio on the frequency of CNT reinforced skew plate. The current results are compared with the related results available in the literature and obtained by different methods. It is shown that reasonable accurate results are obtained for free vibration of nanocomposite plates with less computational effort for higher modes. Several test examples for different plate have been selected to demonstrate the convergence properties, accuracy, and simplicity in numerical implementation of DSC procedures. This approach has verified the accuracy and applicability of DSC method to the class of problem considered in this study. Furthermore, in the numerical examples in the scope of the study, the results obtained with DSC method using a coarser grid are more accurate than the values obtained by finite elements and differential quadrature (DQ) methods. It is also revealed that the method of discrete singular convolution is a promising and potential approach for computational mechanics of nonrectangular plates with nanocomposite reinforced.     

Nonlinear static response of laminated composite plates by discrete singular convolution method

In this paper, large deflection analysis of laminated composite plates is analysed. Nonlinear governing equation for bending based on first-order shear deformation theory (FSDT) in the von Karman sense is presented. These equations have been solved by the method of discrete singular convolution (DSC). Regularized Shannon’s delta (RSD) kernel and Lagrange delta sequence (LDS) kernel are selected as singular convolution to illustrate the present algorithm. The effects of plate aspect ratio, fiber orientation, boundary conditions, thickness-to-side ratio, and applied load on the nonlinear static response of the laminated plate are investigated.     

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.     

Free vibration analysis of thin arbitrarily laminated anisotropic plates using boundary-continuous displacement Fourier approach

A boundary continuous displacement based Fourier series solution to the boundary-value problem of free vibration of an arbitrarily laminated thin rectangular plate is presented. This powerful approach is employed to solve a system of three highly coupled partial differential equations arising from the Kirchhoff hypothesis as applied to an anisotropic laminate, with the SS2-type simply supported boundary conditions prescribed at all four edges. The accuracy of the computed eigenvalues (natural frequencies) is ascertained by studying the convergence characteristics of the lowest seven natural frequencies, and also by comparison with the computed degenerate FEM (finite element methods) results. Other important numerical results presented include variation of the response quantities of interest with geometric and material parameters, such as fiber orientation angle and longitudinal-to-transverse modulus ratio.     

Free vibration analysis of thick laminated plates using a mixed finite element

Abstract

A mixed finite element scheme based on assumed local high‐order displacements is proposed for the free vibration of thick laminated plates. The effects of transverse shear deformation, transverse normal stress and rotary inertia are considered in the formulation. Cross‐ply laminates with simple supports and angle‐ply laminates with clamped edges are presented as examples. The three dimensional elasticity solutions of cross‐ply laminates with simple supports are used to assess the accuracy of the present scheme. The effects of the span‐to‐thickness, aspect and material anisotropy ratio on the fundamental natural frequency are investigated. The present results are compared with the results in the published literature, and agree closely with the 3‐D elasticity solutions.     

Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method

A new version of the differential quadrature method is presented in this paper to overcome the difficulty existing in the ordinary differential quadrature method for applying multi‐boundary conditions in two‐dimensional problems. Since the weighting coefficients of the first derivative are the same as for the ordinary differential quadrature method even with the introduction of multi‐degree‐of‐freedom at the boundary points, the method is easier to extend to two‐ or three‐dimensional problems. A new version of the differential quadrature plate element has been established for demonstration. The essential difference from the existing old version of the differential quadrature plate element is the way the weighting coefficients are determined. The methodology is worked out in detail and some numerical examples are given to show the efficiency of the present method. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite strip buckling and free vibration analysis of stepped rectangular composite laminated plates

A general spline finite strip method is presented which allows the spline knots to be located arbitrarily along the plate strip and also facilitates the use of analytical integration in evaluating strip properties. The development takes place in the contexts of first‐order shear deformation plate theory and of classical plate theory, and encompasses composite laminated material. The prediction of natural frequencies and buckling stresses of stepped rectangular plates is considered using the new approach in which refinement of knot spacings is used local to a step change. The superstrip concept is used as part of an efficient solution procedure. A number of applications demonstrate the validity and practicability of the developed method. Copyright © 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.     

有面内张力和剪力作用的简支各向异性平行四边形板自由振动、屈曲和弯曲的精确解

对简支各向异性平行四边形板在而内张力和剪力作用下自由振动、屈曲和弯曲问题作了全面的研究。利用叠加法和本问题所固有的对称性优点得到了重富立叶级数解。与现有结果的比较验证了此法的精确性。文后给出了多种几何条件下的自由振动、屈曲和弯曲的数值结果。     

Comparative accuracy of DQ and HDQ methods for three‐dimensional vibration analysis of rectangular plates

An accuracy study between the Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) methods for three‐dimensional elasticity solutions of free vibration of rectangular plates is carried out. The solution capability of the DQ and HDQ methods is first studied. Then the numerical performance of both the methods is compared. It is found that the DQ method displays more superior convergence characteristics over the HDQ method for the lower modes of vibration. However, the HDQ method is advantageous over the DQ method for computing higher modes of vibration. It is also discovered that the DQ and HDQ methods produce better convergent solutions than the Finite Element Method (FEM) when a similar number of discrete points/nodes are used. Copyright © 1999 John Wiley & Sons, Ltd.     

Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure

In this article, a new exact closed-form procedure is presented to solve free vibration analysis of functionally graded rectangular thick plates based on the Reddy’s third-order shear deformation plate theory while the plate has two opposite edges simply supported (i.e., Lévy-type rectangular plates). The elasticity modulus and mass density of the plate are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents whereas Poisson’s ratio is constant. Based on the present solution, five governing complicated partial differential equations of motion were exactly solved by introducing the auxiliary and potential functions and using the method of separation of variables. The validity and high accuracy of the present solutions are investigated by comparing some of the present results with their counterparts reported in literature and the 3-D finite element analysis. It is obvious that the present exact solution can accurately predict not only the out of plane, but also the in-plane modes of FG plate. Furthermore, a new eigenfrequency parameter is defined having its special own characteristics. Finally, the effects of boundary conditions, thickness to length ratio, aspect ratio and the power law index on the frequency parameter of the plate are presented.     

Influence of the edge-boundary conditions on three-dimensional free vibrations of isotropic and cross-ply multilayered rectangular plates

This paper analyses the influence of different sets of edge-boundary conditions on the dynamics of freely vibrating isotropic and cross-ply multilayer laminated rectangular plates. The analysis is carried out within the frame of the full three-dimensional theory of elasticity through a formulation which is based on assumed displacements only; this formulation presents its relevant objectives in a unified manner, regardless of the nature of the stacking patterns of the laminated plates (isotropic, single layers or multi-layers). The analytical and/or numerical performance of the formulation is compared to those few results achievable through the exact three-dimensional theory and/or to those few existing results achieved by other researchers through alternative formulations. Convergence analyses are carried out on eigenvalues, displacement and stress fields in order to describe the capability of the formulation when compared to the exact three-dimensional results. The analysis reveals an interesting dependence on the edge-boundary conditions and highlights the need to carry out deeper investigations even though certain classical boundary conditions are taken into account through the most modern electronic computers.     

简支边界条件下超椭圆蜂窝夹芯板的自由振动

基于Reddy三阶剪切板理论,将铝基蜂窝芯层等效为一正交异性层,等效弹性参数由修正后的Gibson公式得出,对简支边界条件下超椭圆蜂窝夹芯板弯曲振动的固有频率进行了理论推导,应用具体算例与实验值和有限元计算结果进行了验证,结果表明本文方法具有较高的计算精度。研究了板宽、板厚和芯层胞元结构参数对简支边界条件下超椭圆蜂窝夹芯板固有频率的影响,并绘出相应的曲线图,该曲线图对超椭圆蜂窝夹芯板的工程应用具有指导意义。     

Static and free vibration analysis of multilayered functionally graded shells and plates using an efficient zigzag theory

Accurate zigzag theory is presented for static and free vibration analysis of multilayered functionally graded material (FGM) cylindrical shells and rectangular plates by approximating inplane displacements as a combination of linear layerwise and cubic global terms. Governing equations of motion are derived using Hamilton’s principle. The theory yields accurate results for displacements, stresses and natural frequencies in simply-supported functionally graded multilayered cylindrical shell panels and rectangular plates. Effect of changing the volume fraction ratio, aspect ratio and thickness of FGM layer between two homogeneous layers are investigated for a number of multilayered shell and plate laminates.     

Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method

This paper mainly Presents free vibration analyses of metal and ceramic functionally graded plates with the local Kriging meshless method. The Kriging technique is employed to construct shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions. The eigenvalue equations of free vibration problems are based on the first-order shear deformation theory and the local Petrov–Galerkin formulation. The cubic spline function is used as the weight function which vanishes on internal boundaries of local quadrature domains and hence simplifies the implementation. Convergence studies are conducted to examine the stability of the present method. Three types of functionally graded plates – square, skew and quadrilateral plates – are considered as numerical examples to demonstrate the versatility of the present method for free vibration analyses.     

Free vibration analysis of anisotropic material structures using the boundary element method

This paper presents a formulation for the analysis of free vibration in anisotropic structures using the boundary element method. The fundamental solution for elastostatic is used and the inertial terms are treated as body forces providing domain integrals. The dual reciprocity boundary element method is used to reduce domain integrals to boundary integrals. Mode shapes and natural frequencies for free vibration of orthotropic structures are obtained and compared with finite element results showing good agreement.