Accurate vibration analysis of skew plates by the new version of the differential quadrature method

An accurate free vibration analysis of skew plates is presented by using the new version of the differential quadrature method (DQM). Eight combinations of simply supported (S), clamped (C) and free (F) boundary conditions are considered. Detailed solution procedures are given and key points for success by using the DQM are emphasized. A way to simplifying the programming in using the DQM is proposed. Convergence study is made for the simply supported skew plate with a large skew angle. Good convergence of frequencies is observed. The DQ results agree very well with the existing first known accurate upper bound solutions, obtained by using Ritz method taking into considerations of the bending stress singularities occurred at corners having obtuse angles. Since slight discrepancy between the DQ data and the known accurate solutions is observed for plates with large skew angles, the DQ results are also compared with data obtained by using finite element method with very fine meshes to verify their accuracy.     

Free vibration analysis of piezoelectric coupled annular plates with variable thickness

This paper presents the free vibration analysis of piezoelectric coupled annular plates with variable thickness on the basis of the Mindlin plate theory. No work has yet been done on piezoelectric laminated plates while the thickness is variable. Two piezoelectric layers are embedded on the upper and lower surfaces of the host plate. The host plate thickness is linearly increased in the radial direction while the piezoelectric layers thicknesses remain constant along the radial direction. Different combinations of three types of boundary conditions i.e. clamped, simply supported, and free end conditions are considered at the inner and outer edges of plate. The Maxwell static electricity equation in piezoelectric layers is satisfied using a quadratic distribution of electric potential along the thickness. The natural frequencies are obtained utilizing a Rayleigh–Ritz energy approach and are validated by comparing with those obtained by finite element analysis. A good compliance is observed between numerical solution and finite element analysis. Convergence study is performed in order to verify the numerical stability of the present method. The effects of different geometrical parameters such as the thickness of piezoelectric layers and the angle of host plate on the natural frequencies of the assembly are investigated.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.     

Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations

This paper presents a nonlinear free vibration analysis of corrugated circular plates with shallow sinusoidal corrugations under uniformly static ambient temperature. Based on the nonlinear bending theory of thin shallow shells, the governing equations for corrugated plates are established from Hamilton’s principle. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude for nonlinear vibration of heated corrugated plates are obtained successfully. The comparison with available published results shows that the numerical analysis here is of good reliability. A detailed parametric study is conducted involving the dependency of nonlinear frequency on the depth and density of corrugations along with the temperature change. Effects of these variables on the trend of nonlinearity are plotted and discussed.     

New benchmark solutions for free vibration of clamped rectangular thick plates and their variants

It is of significance to explore benchmark analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges, because the classic analytic methods are usually invalid for the problems of this category. The main challenge is to find the solutions meeting both the governing higher order partial differential equations (PDEs) and boundary conditions of the plates, i.e., to analytically solve associated complex boundary value problems of PDEs. In this letter, we extend a novel symplectic superposition method to the free vibration problems of clamped rectangular thick plates, with the analytic frequency solutions obtained by a brief set of equations. It is found that the analytic solutions of clamped plates can simply reduce to their variants with any combinations of clamped and simply supported edges via an easy relaxation of boundary conditions. The new results yielded in this letter are not only useful for rapid design of thick plate structures but also provide reliable benchmarks for checking the validity of other new solution methods.     

Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods

The free bending vibration of rotating axially functionally graded (FG) Timoshenko tapered beams (TTB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL). These two methods are capable of modelling any beam whose cross sectional area, moment of inertia and material properties vary along the beam. In order to verify the competency of these two methods, natural frequencies are obtained for problems by considering the effect of material non-homogeneity, taper ratio, shear deformation parameter, rotating speed parameter, hub radius and tip mass. The results are tabulated and compared with the previous published results wherever available.     

An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates

An analytical solution based on a new exact closed form procedure is presented for free vibration analysis of stepped circular and annular FG plates via first order shear deformation plate theory of Mindlin. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. Based on the domain decomposition technique, five highly coupled governing partial differential equations of motion for freely vibrating FG plates were exactly solved by introducing the new potential functions as well as using the method of separation of variables. Several comparison studies were presented by those reported in the literature and the FEM analysis, for various thickness values and combinations of stepped thickness variations of circular/annular FG plates to demonstrate highly stability and accuracy of present exact procedure. The effect of the geometrical and material plate parameters such as step thickness ratios, step locations and the power law index on the natural frequencies of FG plates is investigated.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method

The subject of this article is solving free vibration problems of isotropic and orthotropic rectangular plates with linearly varying thickness along one direction. For the numerical solution to evaluate the frequencies of plates, the method of discrete singular convolution (DSC) is adopted. Frequency parameters are obtained for different types of boundary conditions, taper and aspect ratios. The effect of the mode number is also analyzed. The results obtained by the present numerical method show an excellent agreement with available published results.     

An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates

In this paper, a simple and efficient mixed Ritz-differential quadrature (DQ) method is presented for free vibration and buckling analysis of orthotropic rectangular plates. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQ sample points. The proposed method is suitable for the problem considered due to its simplicity and potential for further development.     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Free vibration characteristics of a functionally graded beam by finite element method

This paper presents the dynamic characteristics of functionally graded beam with material graduation in axially or transversally through the thickness based on the power law. The present model is more effective for replacing the non-uniform geometrical beam with axially or transversally uniform geometrical graded beam. The system of equations of motion is derived by using the principle of virtual work under the assumptions of the Euler–Bernoulli beam theory. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Numerical results are presented in both tabular and graphical forms to figure out the effects of different material distribution, slenderness ratios, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of the beam.     

Improving the differential transform method: A novel technique to obtain the differential transforms of nonlinearities by the Adomian polynomials

Although being powerful, the differential transform method (DTM) yet suffers from a drawback which is the lack of a systematic methodology for derivation of the differential transforms for nonlinear expressions. In the current paper, it is shown that this defect can be overcome with the help of the Adomian polynomials perfectly. The application of the proposed technique in treatment of nonlinear differential equations is well illustrated by a number of examples. In addition, the transformed analogues of some frequent nonlinearities are presented.