A note on elliptical plate vibration modes as a bifurcation from circular plate modes

The exact modes of vibration of a circular plate satisfy the geometrical boundary conditions of uniform elliptical plates in modified polar coordinates. In a previous investigation the exact modes of a clamped circular plate were used as shape functions in the Rayleigh-Ritz method to characterize the vibration of clamped elliptical plates. The mass and stiffness matrices were expressed in closed form and the resulting eigen value problems for the four mode categories of the elliptical plates were solved numerically. In the present investigation the computed variations of the plate frequencies with aspect ratio are used to study the similarities between the elliptical and circular plate modes. It is concluded that as the aspect ratio increases from unity, the axisymmetrical circular plate mode varies as a single elliptical plate mode, whereas the non-axisymmetrical circular plate mode splits up into two distinct elliptical plate modes.     

Free vibration analysis of isotropic and orthotropic triangular plates

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.     

Transverse vibration of completely-free elliptic and circular plates using orthogonal polynomials in the Rayleigh-Ritz method

The Rayleigh-Ritz method using characteristic orthogonal polynomials has been applied to study the problem of transverse vibration of completely-free elliptic and circular plates of uniform thickness. A sequence of successive approximations has been generated to work out the frequencies and mode shapes. The process is continued until at least the first four frequencies converge to five significant digits. It has been found that the results agree completely with the known results for a circular plate. A comparison has also been made with some known experimental and theoretical results for an elliptic plate.     

Frequency solutions for circular plates with internal supports and discontinuous boundaries

The paper presents a study of the free-flexural vibration analysis of circular plates continuous over point supports, partial internal curved supports, and with mixed-edge boundary conditions. An approximate model which combines the advantages of the Rayleigh-Ritz and the Lagrangian multiplier methods is developed for analyzing this class of circular plate problems. The Rayleigh-Ritz method is used to formulate plates with classical boundary conditions, such as free, simply-supported or clamped, while the Lagrangian multiplier method is used to handle plates with point supports, partial internal curved supports and mixed-edge boundary conditions. The admissible pb-2 Ritz function consists of the product of a two-dimensional polynomial and a basic function. The basic function is defined by the product of the equations of the prescribed piecewise-continuous boundary shape each raised to the power of 0, 1 or 2, corresponding to free, simply-supported or clamped edge, respectively. The set of functions automatically satisfies all the kinematic boundary conditions of the plate at the outset. The geometric boundary conditions associated with the internal supports and discontinuous edges are simulated using a sufficient number of closely-spaced point constraints. Numerical results for several selected plate problems are presented to demonstrate the various features and accuracy of the present method.     

Vibration of clamped elliptical plates using exact circular plate modes as shape functions in Rayleigh-Ritz method

The vibration of clamped uniform elliptical plates is investigated using the exact modes of circular plates as shape functions in the Rayleigh-Ritz method. This paper investigates the behaviour of clamped elliptical plates using modified polar coordinates and the exact modes of circular plates in terms of Bessel and modified Bessel functions. The elements of the mass and stiffness matrices are expressed in closed form using Lommel integrals. The accuracy and the convergence of this approach are validated, and numerical results for natural frequencies for various plate aspect ratios are presented.     

带有任意边缘裂纹的自由圆板流固耦合动力特性

提出一种基于Rayleigh-Ritz法,分析带有任意深度、任意角度边缘裂纹的自由圆板流固耦合动力特性的方法,其中,圆板被置于无限大障板中并与单面无限大水域相临。采用同时包含对称和反对称模态振型函数的位移基函数表达带有任意边缘裂纹圆板自由振动的位移;在水体无旋、无粘、不可压假设的基础上,根据Green函数法获得水体的附加质量密度,并基于Ritz过程获得裂纹圆板湿模态对应的固有频率和振型。在与ANSYS数值仿真结果进行对比并验证方法的有效性和精度后,分析水体对带有不同深度和不同角度边缘裂纹的自由圆板动力特性的影响。计算结果表明:水体引起的固有频率降低率整体上随模态阶数的增加而减小,而裂纹深度与角度对固有频率降低率的影响较小;裂纹的出现会增大干、湿振型的差别,某些高阶模态的干、湿振型差别较大。     

Forced axisymmetric response of linearly tapered circular plates

Forced axisymmetric response of a circular plate of linearly varying thickness, based on the classical theory, is analyzed by the eigen-function method. An exact solution for the free vibration mode shapes is obtained by the Frobenius method. Clamped and simply-supported plates subjected to symmetric uniformly distributed and concentrated impulsive ring and point loads are solved as example problems. Numerical results computed for transverse deflection and radial stress are plotted in the figures.     

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.     

Transverse vibration of a circular plate with arbitrary thickness variation

Rayleigh-Ritz method has been employed to obtain approximations to frequencies and mode shapes of circular plates with variable thickness. The boundary is either clamped, simply supported or completely free. The main distinguishing feature of the present investigations is that the thickness approximation is done by measuring thickness at a suitable set of sample points and then using interpolation to get the approximating polynomial. Thus, unlike other methods already available in literature where either linear or quadratic variation of thickness has been examined, here one can have a polynomial of arbitrary degree depending upon the number and locations of the sample points. The results have been tabulated in a large number of cases and three-dimensional mode shapes have been plotted for some selected cases. Comparison has been made with available results. A short tables are also given to depict the rate of convergence with the order of approximation.     

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.     

Fluid bounding effect on natural frequencies of fluid-coupled circular plates

This study deals with the free vibration of two identical circular plates coupled with a bounded or unbounded fluid. An analytical method based on the finite Fourier-Bessel series expansion and Rayleigh-Ritz method is suggested. The proposed method is verified by the finite element analysis using commercial program with a good accuracy. The case of bounded or unbounded fluid is studied for the effect on the vibration characteristics of two circular plates. Also, the effect of gap between the plates on the fluid-coupled natural frequencies is investigated.     

Exact vibration results for stepped circular plates with free edges

A pontoon-type, very large floating structure (VLFS) is often modeled as a huge plate with free edges when performing a hydroelastic analysis under the action of waves. The analysis consists of separating the hydrodynamic analysis from the dynamic response analysis of the VLFS. The deflection of the plate is decomposed into vibration modes where as many higher modes as possible should be used to capture the actual deflection shapes and the stresses. It is generally accepted that finite element method and the Ritz-type energy method fail to model zones with steep gradients which are encountered in, for instance, the stress resultants near the free edges of plates [Journal of Engineering Mechanics 1983;109(2):537–56]. Moreover, the natural boundary conditions are not satisfied completely because they are not enforced a priori [International Journal of Solids and Structures 2001;38:6525–58, Journal of Computational Structural Engineering 2001;1(1):49–57, Journal of Structural Engineering ASCE 2002;128(2):249–57, Computers and Structures 2002:80(2):145–54]. Exact solutions for frequencies, mode shapes and modal stress resultants are thus very important as they provide valuable benchmarks for assessing the convergence, accuracy and validity of numerical results obtained using the finite element method. To this end, we present the exact vibration results for stepped circular plates with free edges. When employed in a hydroelastic analysis, these exact vibration solutions yield accurate deflections and stress resultants (stresses) for circular VLFSs with stepped drafts.     

Transverse vibration of a quarter of an elliptic plate with variable thickness

Rayleigh-Ritz method has been used to find the first three frequencies and mode shapes for free flexural vibration of a plate in the form of a quadrant of an ellipse with linear and quadratic thickness variations. The results for a quarter of a circular plate with uniform thickness have been obtained as a special case and comparison has been made with results already available. By taking various combinations of clamped, simply-supported and completely-free edges we get twenty-seven different cases. Tables are given for the first three frequencies for different values of parameters. Mode shapes have also been plotted in some cases. The convergence of results is indicated by working out approximations of various orders.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Three-dimensional free vibration analysis of perforated super elliptical plates via the p-Ritz method

A three-dimensional free vibration analysis of a perforated plate with rounded corners is presented. A solution method based on the linear, small strain, three-dimensional elasticity theory and the p-Ritz algorithm is employed. This analysis method uses sets of uniquely defined one- and two-dimensional polynomial functions as the trial displacements in the thickness and surface directions to arrive the eigenvalue equation which yields the natural frequencies and mode shapes for the perforated super elliptical plates. The accuracy of these results, if possible, is validated through comparison with the available literature. Parametric investigations on the vibration behaviours of the perforated super-elliptical plates with respect to different thickness ratios, cutout sizes and boundary constraints are examined.     

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.     

带硬涂层圆盘构件的固有振动特性的精确解法

研究了在广义弹性简支边界条件下的具有硬涂层的圆盘构件的自由振动的量纲一固有频率的精确解.首先利用多铁性多层圆盘的解析分析的多层板弹性理论,导出带硬涂层的圆盘结构的状态方程,其中以位移、电势、磁势、应力、电位移和磁感应强度为状态变量.利用有限Hankel变换和传播矩阵法,得到考虑压电和压磁效应的带硬涂层的圆盘的量纲一固有频率的精确解.根据算例结果,比较了压电、压磁两类硬涂层材料在单面涂层、双面涂层和不同涂层厚度的结构配置下的固有频率变化规律.     

Natural frequencies of symmetrically laminated elliptical and circular plates

Elliptical and circular fibre reinforced composite plates are important structural elements in modern engineering structures. Vibration analysis of these elements are of interest to structural designers. The present paper deals with the free transverse vibration analysis of symmetrically laminated solid and annular elliptic and circular plates with several complicating effects.The approach developed is based on the Rayleigh–Ritz method where the deflection of the plate is approximated by a general shape function of polynomial type.The analysis includes several complicating effects, such as the presence of an internal hole, an internal ring support, several concentrated masses and the boundary elastically restrained against rotation and translation.Several examples are solved and some results which correspond to particular cases are compared with existing values in the literature. New results are also presented for cross-ply and angle-ply elliptical and circular laminates with different boundary conditions.The algorithm developed can be applied to a wide range of elastic restraint conditions, to any aspect ratio and to higher modes. The effect of the restraint parameters along the boundary on the natural frequencies for plates with these complicating effects is considered.     

On the use of 2-D orthogonal polynomials in the Rayleigh-Ritz method for flexural vibration of annular sector plates of arbitrary shape

This paper presents a simple, computationally efficient and relatively accurate numerical model for the study of free vibration behaviour of sectorial plates of arbitrary shape. The approach is developed based on the Rayleigh-Ritz method with a set of 2-D orthogonal polynomials as the admissible displacement function where the governing eigenvalue equation is derived by minimizing the energy functional with respect to each of the assumed unknown coefficients. In this paper, several examples are solved and the results obtained are compared with the established solutions to demonstrate the accuracy and flexibility of the present method. New results are also determined for several sectorial plates with different geometrical shapes and combinations of boundary conditions.     

Free vibration analysis of multi-directional functionally graded circular and annular plates

This paper addresses the free vibration of multi-directional functionally graded circular and annular plates using a semianalytical/ numerical method, called state space-based differential quadrature method. Three-dimensional elasticity equations are derived for multi-directional functionally graded plates and a solution is given by the semi-analytical/numerical method. This method gives an analytical solution along the thickness direction, using a state space method and a numerical solution using differential quadrature method. Some numerical examples are presented to show the accuracy and convergence of the method. The most of simulations of the present study have been validated by the existing literature. The non-dimensional frequencies and corresponding displacements mode shapes are obtained. Then the influences of thickness ratio and graded indexes are demonstrated on the non-dimensional natural frequencies.