Free Vibration Analysis Of Plates Research Articles

An Integrated Kirchhoff Element by Galerkin Method for Free Vibration Analysis of Plates on Elastic Foundation

Plates resting on elastic foundations have many engineering applications, especially in the area of structural engineering. Conventionally these foundation systems can be analyzed using regular plate bending element and discrete soil springs. The present work aims an element formulation which is suitable for dynamic analysis of such elastic foundation systems without the use of explicit discrete soil springs. The scope of the work includes free vibration analysis of an isotropic rectangular plate on elastic foundation with various boundary conditions for different thickness and varying properties of foundation. In this paper, finite element modelling has been done for a rectangular isotropic plate by using a four nodded Kirchhoff rectangular element having three degrees of freedom per node, combined with Winkler model of elastic foundation. Here the formulation has been carried out by integrating the properties of the rectangular plate with properties of elastic foundation using Galerkin weighted residual approach instead of the common methods in use like potential energy approach. Numerical modelling has been carried out by developing a MATLAB code and the results of free vibration analysis obtained are in good agreement with the results reported in earlier studies.

Free vibration analysis for plates with arbitrary boundary conditions using a novel spectral-dynamic stiffness method

An exact method for free vibration analysis of plates with arbitrary boundary conditions is presented. This is achieved by integrating the spectral method into the classical dynamic stiffness method. The formulation satisfies the governing differential equation exactly and any arbitrary boundary conditions are satisfied in a series sense. The Wittrick–Williams algorithm is enhanced with several elegant techniques to obtain solutions. The exactness and computational efficiency of the method are demonstrated by comparing results obtained from other methods. Finally, mathematical and physical insights are gained and significant conclusions are drawn for various analytical methods for free vibration analysis of plates.

Free vibration analysis of plate assemblies using the dynamic stiffness method based on the higher order shear deformation theory

This paper deals with the free vibration analysis of isotropic plate assemblies using the dynamic stiffness method (DSM) based on the Reddy’s higher-order shear deformation theory (HSDT). Using the proposed method, the isotropic rectangular plate assemblies of non-uniform thickness and material properties can be analyzed. The proposed model does not have any restrictions regarding the boundary conditions or the frequency limitations. It enables free vibration analysis of both thin and thick plates, making it advantageous in comparison with the conventional finite element method (FEM) regarding the computational cost and the accuracy of the results.Three coupled Euler-Lagrange equations of motion based on the HSDT have been transformed into two uncoupled equations of motion introducing a boundary layer function. The dynamic stiffness matrix for a completely free rectangular plate element has been derived using the superposition and the projection method.The proposed numerical model has been applied in the free vibration analysis of rectangular plate assemblies. Along with the convergence study, the results for natural frequencies have been validated against the existing data from the literature, the previous results from the authors as well as the results obtained by using the finite element software Abaqus. Excellent agreement has been obtained. Finally, a variety of new results is provided as a benchmark for future investigations.

Application of RKP‐FSM in the buckling and free vibration analysis of thin plates with abrupt thickness changes and internal supports

SummaryA description is given of the development and use of the Reproducing Kernel Particle Finite Strip Method for the buckling and flexural vibration analysis of plates with intermediate supports and step thickness changes. The generalized 1‐D shape functions of the Reproducing Kernel Particle Method replace the spline functions in the conventional spline finite strip method in the longitudinal direction. The structure of the generalized Reproducing Kernel Particle Method makes it a suitable tool for dealing with derivative‐type essential boundary conditions, and its introduction in the finite strip method is beneficial for solving buckling and vibration problems for thin plates in which a number of the essential boundary conditions can include the first derivatives of the displacement function. Moreover, the modified corrected collocation method is further developed for the buckling and free vibration analysis of plates with abrupt thickness changes. This provides a versatile and powerful analysis capability which facilitates the analysis of problems including plate structures with abrupt thickness changes of its component plates. The application of the proposed technique for the treatment of discontinuities and the enforcement of the internal support conditions are illustrated with a series of numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin Plates

A new smoothed finite element method (S-FEM) is proposed using hybrid smoothing operations based on nodes and edges of the mesh for static and free vibration analyses of plates governed by the Reissner---Mindlin plate theory. In the present approach, both the node-based smoothed finite element method (NS-FEM) and edge-based smoothed finite element method (ES-FEM) are utilized in a careful designed manner to overcome the shear locking. The formulations use 3-node triangular elements for easy automatic mesh creation, and linear interpolation functions are used for simplicity and robustness. The bending strains field are smoothed by the means of gradient smoothing technique over smoothing domains constructed by element edges, while the shear strains filed is smoothed based on the combination of NS-FEM and ES-FEM with a proper weightage controlled by a coefficient. A simple formula is developed for automatic selection of the coefficient by considering mesh size and thickness of the plate. For easy reference, the present technique is termed as NS+ES-FEM. The numerical examples demonstrate that the proposed method passes the shear-locking test and improves accuracy of the solution.

An edge-based smoothed three-node mindlin plate element (ES-MIN3) for static and free vibration analyses of plates

The edge-based strain smoothing technique is combined with the three-node Mindlin plate element (MIN3) to give a so-called the edge-based smoothed three-node Mindlin plate element (ES-MIN3) for static and free vibration analyses of plates. In the formulation of the ES-MIN3, the MIN3 together with a shear stabilized technique is used to compute the strains and to avoid the shear locking. Then the strain smoothing technique on the smoothing domains associated with edges of elements is used to smooth the strains on the adjacent elements. This strain smoothing technique can provide proper softening effect, which will effectively relieve the stiff behavior of the MIN3 and thus improve significantly the accuracy of solutions. The numerical examples demonstrated that the ESMIN3 is free of shear locking, passes the patch test and shows three superior properties such as: (1) be a strong competitor to others existing plate elements in the static analysis; (2) can give high accurate solutions in free vibration analysis; (3) can provide accurately the values of high frequencies of plates by using only coarse meshes.

Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders

Abstract The paper proposes a comparison between classical two-dimensional (2D) finite elements (FEs) and an exact three-dimensional (3D) solution for the free vibration analysis of one-layered and multilayered isotropic, composite and sandwich plates and cylinders. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the finite element method and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 2D finite element results are obtained by means of a well-known commercial FE code. The differences between 2D FE solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.

Meshless natural neighbor Galerkin method for the bending and vibration analysis of composite plates

In the present work we present a meshless natural neighbor Galerkin method for the bending and vibration analysis of plates and laminates. The method has distinct advantages of geometric flexibility of meshless method. The compact support and the connectivity between the nodes forming the compact support are performed dynamically at the run time using the natural neighbor concept. By this method the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational effort in construction of the shape functions. Smooth non-polynomial type interpolation functions are used for the approximation of inplane and out of plane primary variables. The use of non-polynomial type interpolants has distinct advantage that the order of interpolation can be easily elevated through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background integration cells. The plate formulation is based on first order shear deformation plate theory. The application of natural neighbor Galerkin method formulation has been made for the bending and free vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy of the present numerical method in calculating deflections, stresses and natural frequencies in comparison to the Finite element method, analytical methods and other meshless methods available in the literature.

Experimental and numerical identification of structural modes for engineering education

Development of simple classroom demonstration device and software for visualization of structural normal modes is presented. Device is made of parts of old speaker, controlled with personal computer, where the harmonic motion of solenoid is used as an excitation for beam and plate models. Simple code for finite element free vibration analysis of plates is written in Wolfram Mathematica. Good agreement of results and attractive visual patterns of normal modes attracted attention of students. Results are confirmed using modern modal testing methods. Presented approach is complementary to standard teaching of structural dynamics.

An efficient shear deformation theory for free vibration of functionally graded thick rectangular plates on elastic foundation

Free vibration analysis of plates made of functionally graded materials and resting on elastic foundation is presented by taking into account the effect of transverse shear deformations. The foundation is described by the Pasternak (two-parameter) model. The material properties of the plate are assumed to vary continuously in the thickness direction by a simple power law distribution in terms of the volume fractions of the constituents. The formulation is based on a higher order hyperbolic shear deformation theory. The equation of motion for thick functionally graded plates is obtained through the Hamilton’s principle. The closed form solutions are obtained by using Navier technique and then fundamental frequencies are found by solving the results of eigenvalue problems. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.

Simplified approach of free vibration analysis of plates supported in vicinity of the corners by BEM

Free vibration analysis of Kirchhoff plate by the Boundary Element Method is presented in the paper. The boundary integral equation are derived according to the Bettie theorem. The collocation version of BEM with non-singular approach with one and double collocation points is used. The constant type of element is introduced. Boundary support at selected point is modelled as support in vicinity of point along single boundary element. Introduction Plates supported at points, in vicinity of selected points, for example by the columns, are often used in building structures. The bending problem of plates supported on boundary or rested on internal supports is the classic boundary issue in theory of plates and shells. The columns are often located on plate edges. The first well known work which included strict solutions of these problems is a publication of Nadai [1]. Iguchi [2] considered free vibrations of thin plates supported at the corners. Kączkowski [3] applied double Fourier range of sines to derive a general solution of boundary problem of orthotropic rectangular plates with optional boundary conditions including internal supports located at the plate corners. Timoshenko and Woinowsky-Krieger [4] analysed a static problem of a square plate rested on supports located at the corners by. The solutions as polynominals and range of cosines were assumed. Woźnica [5] solved analytically similar static and dynamic problem using Fourier transformation. A lot of papers present numerical approach, especially the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite Strip Method (FSM), as a useful tool in plate bending analysis. Wide review of the literature is included and cited by Guminiak and Sygulski [6]. The Boundary Element Method (BEM) was proposed as independent numerical tool for applying for engineering applications [7, 8]. Bezine [9] and Stern [10] introduced BEM for analysis of plate bending according to Kirchhoff theory. Vander Weeen [11] proposed fundamental solutions for Reissner plate and used BEM to numerical analysis. Thick plates according to Reissner theory were analysed by Litewka and Sygulski [12] by the BEM. Authors applied fundamental solution derived by Ganowicz [13]. Myślecki [14, 15], Myślecki and Olenkiewicz [16] and Olenkiewicz [17] applied BEM tech-

Free vibration analysis of plates by using hierarchical solid elements regulating the displacement field

Free vibration analysis of plates by using hierarchical solid elements regulating the displacement field

An Improved Meshless Method for the Static and Vibration Analysis of Plates

An application of a meshless technique for the static and vibration analysis of plates is presented in this paper. The method is based on the radial basis functions-QR (RBF-QR) algorithm. Static deformations and free vibration analysis of plates are obtained by Gaussian RBF-QR technique, previously proposed by Fornberg and Piret (2007) for interpolation problems. Circular and square plates are analyzed considering free-edge, simply-supported, and fully-clamped boundary conditions. The presented technique can be applied to any geometry and node sets embedded in the 2D unitary disk. The RBF-QR approach is stable for decreasing values of the shape parameter, corresponding to flat RBFs and the conversion to polar coordinates brings an extra improvement in precision. The method produces highly accurate displacements, natural frequencies, and modes of vibrations.

High accuracy mixed finite element-Ritz formulation for free vibration analysis of plates with general boundary conditions

A high order accurate mixed finite element-Ritz method is introduced and developed to study the vibration problem of plates with general boundary conditions. The finite element method (FEM) with higher order interpolation functions is first used to discretize the spatial partial derivatives with respect to a co-ordinate direction of the plate. The Ritz method is then employed to analogize the resulting system of ordinary differential equations. A novel technique is also presented to exactly satisfy the mixed natural boundary conditions. The proposed method is applied here to solve some benchmark vibration problems of plates including isotropic and anisotropic rectangular plates, variable thickness rectangular plates, multi-span rectangular plates, and skew plates. Comparisons with existing numerical and analytical solutions show that the proposed mixed method can yield highly accurate results for vibration problem of plates involving free edges, free corners and irregular boundaries using a small number of finite elements and Ritz terms.

Free Vibration Analysis of Plates on Winkler Elastic Foundation by Boundary Element Method

Plates are widely used in the machine tool structure. Based on the differential equations of flexural free vibration of a thin plate on Winkler elastic foundation, its dynamic fundamental solutions and the dynamic boundary integral equations are established. The solving methods of the corner point problems are given out. The quadratic elements are used to discretize the boundary integral equations and the receptance mtrix equations of a thin plate on Winkler elastic foundation are established. The fequency scanning method is used to obtain the natural frequencies of the thin plate on Winkler elastic foundation. Case studies showed the effectiveness of the proposed BEM dynamical analysis, which can be also used in the dymical analysis of joints in a machine tool structure.

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates

The cell-based strain smoothing technique is combined with the well-known three-node Mindlin plate element (MIN3) to give a so-called the cell-based smoothed MIN3 (CS-MIN3) for static and free vibration analyses of plates. In the process of formulating the system stiffness matrix of the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized MIN3 is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The numerical examples demonstrated that the CS-MIN3 is free of shear locking, passes the patch test and shows four superior properties such as: (1) be a strong competitor to many existing three-node triangular plate elements in the static analysis, (2) can give high accurate solutions for problems with skew geometries in the static analysis, (3) can give high accurate solutions in free vibration analysis, (4) can provide accurately the values of high frequencies of plates by using only coarse meshes.

The boundedness of Gorman’s Superposition method for free vibration analysis of plates

Gorman’s Superposition method is known as one of the most efficient methods to solve the eigenvalue problems of plates because of its excellent convergence rate. However, there are few published results available that provide sufficient information on its boundedness. Here we have considered the nature of convergence of the eigenvalues for rectangular plates with the following sets of boundary conditions, completely free, fully clamped and cantilever. This paper shows numerically, the boundedness of the Superposition method for undamped vibration problems of rectangular isotropic plates subjected to different boundary conditions. The Superposition method gives upper bound results for eigenvalues of plates if the building blocks used in the Superposition method are subjected to stiffer boundary conditions than those of the original system being modelled. In contrast, the Superposition method yields lower bound results if the boundary conditions of building blocks are more flexible than those of the original system. The results would be useful to estimate the maximum possible error if the other bound can be obtained by another method.

Coupling of h and p finite elements: Application to free vibration analysis of plates with curvilinear plan-forms

This paper presents a method for coupling isoparametric cubic quadrilateral h-elements and straight sided serendipity quadrilateral p-elements. The p-elements are used to model the interior of the domain while the h-elements are used to describe accurately the curved boundaries. At a common side shared by a p-element and an arbitrary number of h-elements, the field variables are minimized in the least square sense with respect to the degrees-of-freedom of the h-elements. This leads to a set of equations which relate the degrees-of-freedom of the coupled elements on the shared side. The method is applied to the calculation of frequencies for plates with curvilinear plan-forms. The effects of shear deformation and rotary inertia are taken into account. The frequencies are obtained for a sectorial plate with simply supported radial edges and free circular edge, an annular sectorial plate with simply supported radial edges and clamped circular edges, and a circular plate with one concentric ring support. Furthermore, new accurate frequencies are given for a fully clamped square plate with a corner cut-out. Constant meshes are used and convergence is sought by increasing progressively the degree p of the interpolating polynomial. The fast convergence and high accuracy of the method are validated through convergence and comparison studies.

A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient

The novelty of this paper is the use of four variable refined plate theory for free vibration analysis of plates made of functionally graded materials with an arbitrary gradient. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents with an arbitrary gradient. The equation of motion for FG rectangular plates is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. In the case of FG clamped plates, the free vibration frequencies are obtained by applying the Ritz method where the four displacement components are assumed as the series of simple algebraic polynomials. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the free vibration behavior of FG plates. Illustrative examples are given also to show the effects of varying gradients, aspect ratios, and thickness to length ratios on the free vibration of the FG plates.

Dynamic stiffness method for exact inplane free vibration analysis of plates and plate assemblies

The inplane free vibration behaviour of plates is investigated using the dynamic stiffness method. Some distinctive modes which went unnoticed in earlier investigations using the dynamic stiffness method have been addressed by revisiting the problem and focusing on the special set of missing solutions. Results are validated extensively both by published results as well as by numerical studies using NASTRAN and ABAQUS. The accuracy of the finite element method for inplane free vibration analysis is assessed and critically examined through the provision of benchmark solutions. Some representative modes that are missed by well-established dynamic-stiffness-based computer programs are presented. The inplane dynamic stiffness matrix presented is of great importance when combined with the out of plane matrix in order to obtain the closed-form solution for free vibration analysis of structures with complex geometries.