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.     

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.     

On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. The variation in material properties through the thickness of the plate introduces a coupling between in-plane and transverse displacements, the coupling is important in the vibration of functionally graded plates (FGPs), but none have produced an exact closed-form solution for the in-plane as well as transverse vibrations of smart circular/annular FGPs. Therefore, this paper develops an exact closed-form solution for the free vibration of piezoelectric coupled thick circular/annular FGPs subjected to different boundary conditions on the basis of the Mindlin’s first-order shear deformation theory. Through the comparison of present results with those available, the accuracy of the present method was verified. The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic. Due to the inherent features of the present solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future.     

用于变厚度圆盘振动分析的动态环形元

本文利用动态位移的回转周期变化规律,将变厚度圆板弯曲振动的二维问题化为一维形式.并将环形单元的位移函数以频率的升幂形式展开,作为动态形状函数,进而推导了动态环形元的质量和刚度矩阵及其一次动态修正矩阵,并以简洁的形式给出.最后作为本文方法的直接应用,计算了叶盘组合件的固有特性与实验值及其它方法进行了比较,说明动态的变厚度环板元的优越性.     

Dynamic analysis of multi-directional functionally graded annular plates

Dynamic analysis of multi-directional functionally graded annular plates is achieved in this paper using a semi-analytical numerical method entitled the state space-based differential quadrature method. Based on the three-dimensional elastic theory and assuming the material properties having an exponent-law variation along the thickness, radial direction or both directions, the frequency equations of free vibration of multi-directional functionally graded annular plates are derived under various boundary conditions. Numerical examples are presented to validate the approach and the superiority of this method is also demonstrated. Then free vibration of functionally graded annular plates is studied for different variations of material properties along the thickness, radial direction and both directions, respectively. And the influences of the material property graded variations on the dynamic behavior are also investigated. The multi-directional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material.     

On natural frequencies of transversely isotropic circular plates

The paper discusses the impact of the material properties of transversely isotropic circular plates on its natural frequencies. Two refined theories of plates have been used to analyze the free vibration behavior of homogeneous plates. Both theories take into account normal and rotary inertias. Fundamental frequencies for plates with radial inhomogeneity have been obtained with the help of finite element package Comsol Multiphysics 5.0. It has been shown that the inhomogeneity of the plate have a profound impact on the first (lowest) frequency of the plate, while the plate orthotropy has a greater influence on the second and higher vibration mode [2] (Fig. 1, Table 1).     

Transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness

An analysis of the transverse vibration of nonhomogeneous orthotropic viscoelastic circular plates of parabolically varying thickness in the radial direction is presented. The thickness of a circular plate varies parabolically in a radial direction. For nonhomogeneity of the circular plate material, density is assumed to vary linearly in a radial direction. This paper used the method of separation of variables in solving the governing differential equation. In this paper, an approximate but quite convenient frequency equation is derived by using the Rayleigh–Ritz technique with a two‐term deflection function. Deflection, time period and logarithmic decrement for the first two modes of vibration are computed for the nonhomogeneous orthotropic viscoelastic circular plates of varying parabolic thickness with clamped edge conditions for various values of nonhomogeneity constants and taper constants and these are shown in tabular form for the Voigt–Kelvin model. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.     

3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates

Within the framework of three-dimensional elasticity theory, this paper investigates the axisymmetric bending of novel functionally graded polymer nanocomposite circular and annular plates reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite circular and annular plates under a combined action of a uniformly distributed transverse load and a through-thickness steady temperature field. Three different distribution patterns of GPLs within the polymer matrix are considered. The present analytical solutions are validated through comparisons against those available in open literature for the reduced cases. A parametric study is conducted to examine the effects of GPL weight fraction, distribution pattern, plate thickness to radius ratio, and boundary conditions on the stress and deformation fields of the plate. The results show that GPL nanofillers with a low content can have a significant reinforcing effect on the bending behavior of the thermo-mechanically loaded plate.     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

New variational-asymptotic formulations for 3-D stress analyses of laminated composite shells with a circular hole

A new approach for three-dimensional stress analyses in composite cylindrical shells is presented. The method of composite expansions along with Hellinger-Reissner variational formulation is employed to derive the interior and edge layer problems for high order approximations. Classical assumptions have been justified and new approximations have been established. These formulations are directed especially towards, new high integrity mixed-hybrid finite element schemes. The expository examples chosen are of cross-ply and angle-ply laminated shells. The circumferential location of the delamination failure initiation, for angle-ply laminates containing a circular hole, is within a sector located symmetrically around the perpendicular direction to the applied load.     

Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry

Many rectangular plate elements developed in the history of finite element method (FEM) have displayed excellent numerical properties, yet their applications have been limited due to inability to conform to the arbitrary geometry of plates and shells. Numerical manifold method (NMM), considered to be a generalization of FEM, can easily solve this issue by viewing a mesh made up of rectangular elements as mathematical cover. In this study, ACM element (Adini and Clough element from A. Adini, R.W. Clough, Analysis of plate bending by the finite element method, University of California, 1960), a typical rectangular plate element is first integrated in the framework of NMM. Then, vibration analysis of arbitrary shaped thin plates is conducted employing the tailored NMM. Using the definition of integral of scalar functions on manifolds, we developed a mathematically rigorous mass lumping scheme for creating a symmetric and positive definite lumped mass matrix that is easy to inverse. A series of numerical experiments have been studied and analyzed, including free and forced vibration of thin plates with various shapes, validating the proposed mass lumping scheme can supersede the consistent mass formulation in those cases.     

Free vibration of polar orthotropic circular plates of quadratically varying thickness resting on elastic foundation

Classical plate theory is used to study free vibration of polar orthotropic circular plates of quadratically varying thickness resting on Winkler elastic foundation. Boundary characteristic orthonormal polynomials are used in the Rayleigh–Ritz method to analyze the plate for free, simply supported and clamped edge conditions. The numerical convergence of the method is tested and comparisons are made in particular case with the results already available in the literature. First twelve frequencies for various values of parameters describing the elastic foundation, thickness profile and orthotropy of the plate are given in tabular form. Two dimensional graphs for nodal lines and three dimensional graphs for mode shapes are plotted.     

Uncovering the finite difference model equivalent to Hencky bar-net model for axisymmetric bending of circular and annular plates

As there are a few finite difference models in the literature for axisymmetric bending of plates, only one of these models is equivalent to the Hencky bar-net model (HBM) that comprises a finite number of rigid circular arcs and straight radial segments joined by frictionless hinges with elastic rotational springs. This paper is concerned with uncovering the one finite difference model (FDM) that is equivalent to the HBM. Based on the energy formulation, the governing equation for HBM is derived and it will be used to identify the FDM that has the same discrete set of equations. By using this equivalency between the HBM and the identified FDM, the expressions of edge spring stiffnesses of HBM are derived for various boundary conditions. As illustrative examples, the HBM is used to solve the bending problems of circular plates under uniformly and linearly increasing distributed loads. The analytical solutions of HBM avoids the singularity problem faced in FDM at the plate center. This paper also presents some benchmark bending solutions for annular plates with and without an internal ring support for different boundary restraints by using the HBM.     

The postbuckling analysis of laminated circular plate with elliptic delamination

Based on the Von Karman plate theory, considering the effect of transverse shear deformation, and using the method of the dissociated three regions, the postbuckling governing equations for the axisymmetric laminated circular plates with elliptical delamination are derived. By using the orthogonal point collocation method, the governing equations, boundary conditions and continuity conditions are transformed into a group of nonlinear algebraically equation and the equations are solved with the alternative method. In the numerical examples, the effects of various elliptical in shape, delamination depth and different material properties on buckling and postbuckling of the laminated circular plates are discussed and the numerical results are compared with available data.     

Asymmetric free vibration of circular plate in contact with incompressible fluid

Vibration of circular plates in contact with fluid has extensive applications in the industry. This paper derives added mass and frequencies for asymmetric free vibration of coupled system including clamped circular plate in contact with incompressible bounded fluid. Considering small oscillations induced by the plate vibration in the incompressible and inviscid fluid, velocity potential function is used to describe the fluid motion. Derivation uses Kirchoff’s thin plate theory. Two approaches are used to derive the free vibration frequency of the system. The solutions include an analytical solution employing Fourier–Bessel series and a variational formulation applied simultaneously on the plate and fluid. Strong correlation is found between free vibration frequencies of the two solutions. Finally the effect of fluid depth on the added mass and free vibration frequencies of the coupled system is investigated.     

Frequency-dependent vibration analysis of symmetric cross-ply laminated plate of Levy-type by spectral element and finite strip procedures

This research describes spectral finite element formulation for vibration analysis of rectangular symmetric cross-ply laminated composite plates of Levy-type based on classical lamination plate theory (CLPT). Formulation based on SFEM includes partial differential equations of motion, spectral displacement field, dynamic shape functions, and spectral element stiffness matrix (SESM). In this paper, vibration analysis of composite plate is investigated in two sections: free vibrations and forced vibrations. In free vibrations, natural frequencies are calculated for different Young’s moduli ratios and boundary conditions. In forced vibrations, plate vibrations are investigated under high-frequency concentrated impulsive loads. The resulting responses due to spectral element formulation are compared with those of (time-domain) finite element and analytical formulations, whenever available. The results demonstrate the superiority of SFEM with respect to FEM, in reducing computational burden, simultaneously increasing numerical accuracy, specifically for excitations of high-frequency content. By reducing the time duration of impulsive loads, and consequently increasing the modal contribution of higher modes in the transient response of plate, the accuracy of FEM responses decreases substantially accompanied with a high volume of computations, while the accuracy of the SFEM response results is very high and simultaneously, with a low computational burden. Practically, SFEM follows very closely exact analytical solutions.     

均布载荷作用下夹层圆板的非线性振动

给出了均布载荷作用下夹层圆板的大幅度振动方程.按假设的时间模态导出了该问题的非线性耦合的代数和微分特征方程组,并利用修正迭代法求出了该方程组的近似解析解,得到了周边固定夹层圆板振动的幅频-载荷特征关系.讨论了载荷对非线性振动性态的影响.     

环形和圆形薄板在各种支承条件下的非对称非线性弯曲问题(Ⅰ)

本文研究环形和圆形薄板在各种支承条件下的非对称非线性弯曲问题.应用[7]中提出的摄动方法导出一致有效的渐近解.