Eigenvalue analysis of rectangular mindlin plates by chebyshev pseudospectral method

A study of free vibration of rectangular Mindlin plates is presented. The analysis is based on the Chebyshev pseudospectral method, which uses test functions that satisfy the boundary conditions as basis functions. The result shows that rapid convergence and accuracy as well as the conceptual simplicity are achieved when the pseudospectral method is applied to the solution of eigenvalue problems. Numerical examples of rectangular Mindlin plates with clamped and simply supported boundary conditions are provided for various aspect ratios and thickness-to-length ratios.     

Free vibration analysis of spherical caps by the pseudospectral method

The pseudospectral method is applied to the axisymmetric and asymmetric free vibration analysis of spherical caps. The displacements and the rotations are expressed by Chebyshev polynomials and Fourier series, and the collocated equations of motion are obtained in terms of the circumferential wave number. Numerical examples are provided for clamped, hinged and free boundary conditions. The results show good agreement with those of existing literature. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from University of Michigan in 1992 and joined Dept. of Mechano-Informatics of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

The application of the pseudospectral method to the analysis of helical springs of arbitrary shape

The pseudospectral method is applied to the static analysis of helical springs of arbitrary shape. The displacements and the rotations are approximated by series expansions of Chebyshev polynomials. The entire domain is considered as a single element and the governing equations are collocated to yield the system of algebraic equations. The boundary conditions are considered as the constraints, and the set of equations is condensed so that the number of degrees of freedom of the problem matches the total number of the expansion coefficients. Displacements and rotations are computed for noncylindrical helical springs as well as cylindrical helical springs, and parameters that affect the convergence of the solution are discussed. This paper was recommended for publication in revised form by Associate Editor Jeong Sam Han Jinhee Lee received B. S. and M. S. degrees from Seoul National Uni-versity and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from University of Michigan in 1992 and joined Dept. of Mechano-Informatics of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

Application of the chebyshev-fourier pseudospectral method to the eigenvalue analysis of circular mindlin plates with free boundary conditions

An eigenvalue analysis of the circular Mindlin plates with free boundary conditions is presented. The analysis is based on the Chebyshev-Fourier pseudospectral method. Even though the eigenvalues of lower vibration modes tend to convergence more slowly than those of higher vibration modes, the eigenvalues converge for sufficiently fine pseudospectral grid resolutions. The eigenvalues of the axisymmetric modes are computed separately. Numerical results are provided for different grid resolutions and for different thickness-to-radius ratios.     

Vibration and buckling characteristics of weld-bonded rectangular plates using the flexibility function approach

Structures with a combination of spot welds and adhesive bonding, often referred to as weld-bonded structures, are likely to see increasing usage in automotive and other engineering structures. The present study considers a representative weld-bonded rectangular plate having simple supports on two opposite edges and weld-bonded support conditions with periodic spot welds along the other two edges. The study shows that the flexibility function approach for modeling free edges with point supports [Bapat AV, Venkatramani N, Suryanarayan S. Simulation of classical edge conditions by finite elastic restraints in the vibration analysis of plates. Journal of Sound and Vibration 1988;120(1):127–40; Bapat AV, Venkatramani N, Suryanarayan S. A new approach for the representation of a point support in the analysis of plates. Journal of Sound and Vibration 1988;120(1):107–25; Bapat AV, Venkatramani N, Suryanarayan S. The use of flexibility functions with negative domains in the vibration analysis of asymmetrically point-supported rectangular plates. Journal of Sound and Vibration 1988;124(3):555–76; Bapat AV, Suryanarayan S. Free vibrations of periodically point-supported rectangular plates. Journal of Sound and Vibration 1989;132(3):491–509; Bapat AV, Suryanarayan S. The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges. Journal of Sound and Vibration 1989;128(2):203–33; Bapat AV, Suryanarayan S. Free vibrations of rectangular plates with interior point supports. Journal of Sound and Vibration 1989;134(2):291–313; Bapat AV, Suryanarayan S. Importance of satisfaction of point-support compatibility conditions in the simulation of point supports by the flexibility function approach. Journal of Sound and Vibration 1990;137(2):191–207; Bapat AV, Suryanarayan S. A fictitious foundation approach to vibration analysis of plates with interior point. Journal of Sound and Vibration 1992;155(2):325–41; Bapat AV, Suryanarayan S. A theoretical basis for the experimental realization of boundary conditions in the vibration analysis of plates. Journal of Sound and Vibration 1993;163(3):463–78], used in the direct series solution of the governing differential equations, can be employed very effectively to study the vibration and buckling characteristics of the weld-bonded rectangular plates. This is done by using a flexibility function constructed in terms of Fourier components to model the weld-bonded edge that represents the finite uniform flexibility of the adhesively bonded segment of the weld-bonded edge along with zero flexibility at the spot welds modeled as discrete point supports. A detailed convergence study shows that by a proper choice of the number of terms used to represent the flexibility function and the number of terms in the Levy sine series for the solution of the plate displacement, accurate results can be obtained for vibration and buckling characteristics. This paper also includes a parametric study undertaken to show the effect of plate geometry, number of spot welds and adhesive joint parameters. The paper also discusses how such parametric studies can be of use to the designer in arriving at an optimal joint configuration of weld-bonded rectangular plates from linear elastic buckling and free vibration considerations.     

Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method

The main objective of this study is to give a numerical solution of three-dimensional analysis of thick rectangular plates. The analysis uses discrete singular convolution (DSC) method. Free vibration, bending and buckling of rectangular plates have been studied in this paper. Regularized Shannon's delta (RSD) kernel is selected as singular convolution to illustrate the present algorithm. In the proposed approach, the derivatives in both the governing equations and the boundary conditions are discretized by the method of DSC. The obtanied results are compared with those of other numerical methods. It is found that the convergence of the DSC approach is very good and the results agree well with those obtained by other researchers.     

Three-dimensional static solutions of rectangular plates by variant differential quadrature method

An endeavor to exploit three-dimensional elasticity solutions for bending and buckling of rectangular plates via the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods is performed. Unlike other works, the priority of this paper is to examine the computational characteristics of the two methods; therefore, we focus our studies only on the simply supported and clamped rectangular plates. To start with, we first outline the basic equations and boundary conditions describing the bending and buckling of rectangular plates followed by normalizing and discretizing them according to the DQ and HDQ algorithms. The resulting algebraic equation systems are then solved to obtain the solutions. Based on these solutions, the computational characteristics of the DQ and HDQ methods are investigated in terms of their numerical performances. It is found that the DQ method displays obvious superior convergence characteristics over the HDQ method for the three-dimensional static analysis of rectangular plates.     

Nonlinear flexural vibration of rectangular moderately thick plates and sandwich plates

Nonlinear flexural vibration is investigated for rectangular Reissner moderately thick plates and sandwich plates. The fundamental equations and boundary conditions are expressed in unified dimensionless form for rectangular moderately thick plates and sandwich plates. Highly accurate solutions of series form with many different movable and immovable boundary conditions, especially with unsymmetrical boundary conditions, are obtained by means of the method of harmonic balance and by developing a new technique of mixed Fourier series in nonlinear analysis. The nonlinear partial differential equations are reduced to an infinite set of simultaneous nonlinear algebraic equations, which are truncated in numerical computations. Solutions of the nonlinear fundamental frequency of rectangular plates are obtained by iteration. The multimode approach includes not only the influences of transverse shearing deformation and rotatory inertia, but also the coupling effect of vibrating modes on the nonlinear fundamental frequency. The present solutions are satisfactory in comparison with other available results.     

Analytical solutions for bending analysis of rectangular laminated plates with arbitrary lamination and boundary conditions

The intent of the present study is to employ the extended Kantorovich method for semi-analytical solutions of laminated composite plates with arbitrary lamination and boundary conditions subjected to transverse loads. The method based on separation of spatial variables of displacement field components. Within the displacement field of a first-order shear deformation theory, a laminated plate theory is developed. Using the principle of minimum total potential energy, two systems of coupled ordinary differential equations with constant coefficients are obtained. The equations are solved analytically by using the state-space approach. The results obtained are compared with the Levy-type solutions of cross-ply and antisymmetric angle-ply laminates with various admissible boundary conditions to verify the validity and accuracy of the present theory. Also, for other laminations and boundary conditions that there exist no Levy-type solutions the present results are compared with those obtained by other investigators and finite element method. It is found that the present results have very good agreements with those obtained by other methods. This paper was recommended for publication in revised form by Associate Editor Heoung Jae Chun Ali Mohammad Naserian Nik received his M.S. in Mechanical Engineering from Ferdowsi University of Mashhad, Iran, in 2006. Currently, he is a doctoral student at the Department of Mechanical Engineering, Ferdowsi University of Mashhad, Iran. His research interests are in the area of computational mechanics and nanobiotechnology. Masoud Tahani is currently an Associate Professor at the Department of Mechanical Engineering at Ferdowsi University of Mashhad, Iran. He received his B.S. in Mechanical Engineering from Ferdowsi University of Mashhad, Iran, in 1995. He then received his M.S. and Ph.D. degrees from Sharif University of Technology, Iran, in 1997 and 2003, respectively. Dr. Tahani’s research interests include design of structures using advanced composites, mechanics of anisotropic materials, smart materials and structures, mechanics of plates and shells and biomechanics.     

Soft-spring nonlinear vibrations of antisymmetrically laminated rectangular plates

This paper deals with the effects of initial geometric imperfections and in-plane boundary conditions on the large-amplitude vibration behavior of angle- and cross-ply rectangular thin plates. It is found that the presence of imperfection amplitudes of the order of only half the total laminated-plate thickness may significantly raise the vibration frequencies and change the large-amplitude vibration behavior from the well-known hard-spring to soft-spring behavior. The effects of fibre angles and bending-stretching coupling for angle-ply plates and Young's moduli ratios and number of layers for antisymmetric cross-ply plates are examined.     

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.     

Transverse vibration of rectangular plates subjected to inplane forces by a difference based variational approach

Free vibration characteristics of rectangular plates subjected to inplane loads have been studied using the variational finite difference method. The total energy of free vibration of the system is discretized by replacing the derivative terms by their finite difference equivalents and energy minimization technique is used to obtain a typical eigenvalue problem. Vibration frequencies for various modes for plates subjected to inplane normal loads, pure shear and their combination have been determined for different aspect ratios and edge conditions. It has been observed that the effect of inplane loads on vibration frequencies is more pronounced in the case of plates having similar modes for vibration and buckling.     

Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

FSDPT based study for vibration analysis of piezoelectric coupled annular FGM plate

The vibration behavior of a piezoelectrically actuated thick functionally graded (FG) annular plate is studied based on first order shear deformation plate theory (FSDPT). A consistent formulation that satisfies the Maxwell static electricity equation is presented so that the full coupling effect of the piezoelectric layer on the dynamic characteristics of the annular FG plate can be estimated based on the free vibration results. The differential equations of motion are solved analytically for various boundary conditions of the plate. The analytical solutions are derived and validated by comparing the obtained resonant frequencies of the composite plate with those of an isotropic core plate. As a special case, assuming that the material composition of core plate varies continuously in the direction of the thickness according to a power law distribution, a comprehensive study is conducted to show the influence of functionally graded index on the vibration behavior of smart structure. Also, the good agreement between the results of this paper and those of the finite element (FE) analyses validates the presented approach. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Farzad Ebrahimi received his B.S. and M.S. degree in Mechanical Engineering from University of Tehran, Iran. He is currently working on his Ph.D. thesis under the title of “Vibration analysis of smart functionally graded plates” at Smart Materials and Structures Lab in Faculty of Mechanical Engineering of the University of Tehran. His research interests include vibration analysis of plates and shells, smart materials and structures and functionally graded materials.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. 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. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Effect of aspect ratio on large amplitude free vibrations of simply supported and clamped rectangular Mindlin plates using coupled displacement field method

We propose a novel method, known as Coupled displacement field (CDF) method, an alternative to study large amplitude free vibration behavior of moderately thick rectangular plates. An admissible trial function was assumed for one of the variables, say, the total rotations (in both X, Y directions). The function for lateral displacement field is derived in terms of the total rotations with the help of coupling equations, where the two independent variables become dependent on one another. This method makes use of the energy formulation, where it contains only half the number of undetermined coefficients when compared with conventional Rayleigh-Ritz method. The vibration problem is simplified significantly due to the reduction in number of undetermined coefficients. The frequency-amplitude relationship for the moderately thick rectangular plates with various aspect ratios for all edges simply supported and clamped boundary conditions was obtained. Closed form expressions for linear and nonlinear fundamental frequency parameters were derived.     

Analytical buckling solutions for mindlin plates involving free edges

This paper investigates the buckling behaviour of rectangular Mindlin plates having two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The proper boundary conditions at free edges subjected to in-plane loads have been examined. The buckling analysis is performed by applying the concept of state space to the Levy-type solution method to obtain the closed-form critical loads from the governing differential equations. The results, where possible, are compared with existing solutions to verify the validity of the solution method. The differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are reported. Several design charts representing the essential features of the critical load characteristics of rectangular plates with two opposite edges simply supported at least one free edge are obtained. The critical loads can be determined from the design charts without difficulty.     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.