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 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.     

Power performance of circular piezoelectric diaphragm generators

Energy generation performance of a piezoelectric generator depends mainly on several elements such as the structural style, boundary conditions, geometry parameters, materials, vibration-source frequency, and external load. To obtain the optimal energy-harvesting device, the Raleigh method is used to establish the analysis model of circular piezoelectric composite diaphragms. Simply supported and clamped boundary conditions were considered. The relationships between the output power and the structural parameters of piezoelectric composite diaphragms, and the external load resistance and frequency were shown. Given the correlative material parameters and boundary conditions, the output power, using structural parameters, external load, or vibrating frequency as variables, can be calculated. Simulation results show that there are optimal structural parameters and load for a composite diaphragm to achieve the maximum output power. A piezoelectric diaphragm generator with given dimensions tends to achieve higher output power under clamped boundary conditions than that under simply supported boundary conditions.     

Frequencies of circular plate with concentric ring and elastic edge support

Exact solutions for the flexural vibrations of circular plates having elastic edge conditions along with rigid concentric ring support have been presented in this paper. Values of frequency parameter for the considered circular plate are computed for different sets of values of elastic rotational and translation restraints and the radius of internal rigid ring support. The results for the first three modes of plate vibrations are computed and are presented in tabular form. The effects of rotational and linear restraints and the radius of the rigid ring support on the vibration behavior of circular plates are studied over a wide range of non-dimensional parametric values. The values of the exact frequency parameter presented in this paper for varying values of restraint parameters and the radius of the rigid ring support can better serve in design and as benchmark solutions to validate the numerical methods obtained by using other methods of solution.     

Power performance of circular piezoelectric diaphragm generators

Energy generation performance of a piezoelectric generator depends mainly on several elements such as the structural style, boundary conditions, geometry parameters, materials, vibration-source frequency, and external load. To obtain the optimal energy-harvesting device, the Raleigh method is used to establish the analysis model of circular piezoelectric composite diaphragms. Simply supported and clamped boundary conditions were considered. The relationships between the output power and the structural parameters of piezoelectric composite diaphragms, and the external load resistance and frequency were shown. Given the correlative material parameters and boundary conditions, the output power, using structural parameters, external load, or vibrating frequency as variables, can be calculated. Simulation results show that there are optimal structural parameters and load for a composite diaphragm to achieve the maximum output power. A piezoelectric diaphragm generator with given dimensions tends to achieve higher output power under clamped boundary conditions than that under simply supported boundary conditions.     

A simple finite element analysis of axisymmetric vibration of annular and circular plates

The axisymmetric vibration of annular and circular plates, isotropic or polar orthotropic, is analysed by an axisymmetric finite element. The features of the present study are that: (i) the formulation of the finite element is based on elasticity theory and has no assumptions as in the conventional plate theory-based analysis, yet is still simple and ready for use, and (ii) the boundary conditions are satisfied exactly and the significant effect of boundary conditions on the vibration frequency is demonstrated. Comparisons with alternative solutions show the accuracy of the present approach and the inadequacy of conventional methods in dealing with the vibration of annular and circular plates with simply supported boundary conditions.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. 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. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution

This paper introduces the discrete singular convolution algorithm for vibration analysis of rectangular plates with mixed boundary conditions. A unified scheme is proposed for the treatment of simply supported, clamped and transversely supported (with nonuniform elastic rotational restraint) boundary conditions. The robustness and reliability of the present approach are tested by a number of numerical experiments. All results agree well with those in the literature.     

A novel finite-element method for use in classical elastic plate problems

We present a novel finite-element method for solving problems in the bending of elastic plates. The basic matrix for an element has mixed force and displacement variables. The coefficients of the matrix are derived without recourse to extremum principles, by the application of conventional structural theory to a set of simple sub-problems or loading cases. Inter-connection of elements by their mid-side nodes is easy, and force or displacement boundary conditions may be imposed directly and easily. The method gives good results, and all force and displacement quantities, including boundary reactions, are readily accessible. The rectangular element here presented is one of a family developed by Woodhead.     

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.     

结构体面力重构的边界元逆分析法研究

结构体的边界条件未能充分给出的情况下,用常规边界元法很难作出正确的结构分析。本文利用边界元法与非线性优化技术相结合,根据弹性体内或边界上的位移、应力等附加信息,建立了结构体面力重构的边界元逆分析法的基本模型,算例证明了该理论的有效性。     

Analysis of the free vibration of rectangular plates with central cut-outs using the discrete Ritz method

We present an analysis of the free vibration of plates with internal discontinuities due to central cut-outs. A numerical formulation for a basic L-shaped element which is divided into appropriate sub-domains that are dependent upon the location of the cut-out is used as the basic building element. Trial functions formed to satisfy certain boundary conditions are employed to define the transverse deflection of each sub-domain. Mathematical treatments in terms of the continuities in displacement, slope, moment, and higher derivatives between the adjacent sub-domains are enforced at the interconnecting edges. The energy functional results, from the proper assembly of the coupled strain and kinetic energy contributions of each sub-domain, are minimized via the Ritz procedure to extract the vibration frequencies and mode shapes of the plates. The procedures are demonstrated by considering plates with central cut-outs that are subjected to two types of boundary conditions.     

Denting of internally pressurized tubes under lateral loads

The present work examines the structural response of tubular members subjected to lateral quasi-static loading, imposed by wedge-shaped denting devices, in the presence of internal pressure. First, tubes are modeled with shell finite elements, accounting for geometric and material nonlinearities, and the numerical results are in good agreement with available denting test data from internally pressurized pipes. Using the finite element tools, a parametric study is conducted and load-deflection curves are obtained for different levels of pressure, for various wedge shapes and for different types of boundary conditions. It is found that the presence of internal pressure increases significantly the denting force. The effects of yield anisotropy on the denting resistance are also examined. A simplified three-dimensional analytical model is also developed, based on rate-of-energy balance, which yields closed-form expressions for the denting force and the corresponding denting length. The model, introduced elsewhere for non-pressurized tubes, is enhanced to include the pressure effects, accounting for different types of pipe end conditions, as well as the effects of plastic anisotropy. The analytical solution compares very well with finite element results, and illustrates tube denting response in a clear and elegant manner.     

Vibration analysis of piezoelectric FGM sensors using an accurate method

In this study, free vibration analysis of moderately thick smart FG annular/circular plates with different boundary conditions is presented on the basis of the Mindlin plate theory. This structure comprised a host FG plate and two bonded piezoelectric layers. Piezoelectric layers are open circuit therefore this plate can be used as a sensor. According to power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate while Poisson's ratio is set to be constant. Using Hamilton's principle and Maxwell electrostatic equation yields six complex coupled equations which are solved via an exact closed-form method. The accuracy of the frequencies is verified by the available literature, finite element method (FEM) and the Kirchhoff theory. The effects of plate parameters like boundary condition and gradient index are investigated and significance of coupling between in-plane and transverse displacements on the resonant frequency is proved.     

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.     

Finite element correction matrices in metal forming analysis (with application to hydrostatic bulging of a circular sheet)

This paper presents a condensed review of finite element applications in metal forming processes. Difficulties in large strain-elastoplastic analysis of these processes are identified. Correction techniques for potential sources of errors, such as lack of nodal equilibrium, violation of yield conditions, overstiff performance of finite elements, and improper formulation of configuration-dependent problems, are discussed. A main feature of the paper is the account of mixed boundary data in the updated Lagrangian formulation variational procedure. The code developed is applied to the process of the hydrostatic bulging of a circular sheet clamped at its periphery. This is based on a convenient explicit form of the hydrostatic-pressure load correction stiffness matrix for isoparametric and allied types of elements.     

Analysis of rectangular laminated composite plates via FSDT meshless method

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates.     

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.     

Free vibration analysis of rectangular plates with internal columns and uniform elastic edge supports by pb-2 Ritz method

Free vibration analysis of rectangular plates with internal columns and elastic edge supports is presented using the powerful pb-2 Ritz method. Reddy's third order shear deformation plate theory is employed. The versatile pb-2 Ritz functions defined by the product of a two-dimensional polynomial and a basic function are taken as the admissible functions. Substituting these displacement functions into the energy functional and minimizing the total energy by differentiation, leads to a typical eigenvalue problem, which is solved by a standard eigenvalue solver. Stiffness and mass matrices are numerically integrated over the plate using the Gaussian quadrature. The accuracy and efficiency of the proposed method are demonstrated through several numerical examples by comparison and convergency studies. Many numerical results for reasonable natural frequency parameters of rectangular plates with different combinations of elastic boundary conditions and column supports at any locations are presented, which can be used as a benchmark for future studies in this area.     

XFEM schemes for level set based structural optimization

In this paper, some elegant extended finite element method (XFEM) schemes for level set method structural optimization are proposed. Firstly, two- dimension (2D) and three-dimension (3D) XFEM schemes with partition integral method are developed and numerical examples are employed to evaluate their accuracy, which indicate that an accurate analysis result can be obtained on the structural boundary. Furthermore, the methods for improving the computational accuracy and efficiency of XFEM are studied, which include the XFEM integral scheme without quadrature sub-cells and higher order element XFEM scheme. Numerical examples show that the XFEM scheme without quadrature sub-cells can yield similar accuracy of structural analysis while prominently reducing the time cost and that higher order XFEM elements can improve the computational accuracy of structural analysis in the boundary elements, but the time cost is increasing. Therefore, the balance of time cost between FE system scale and the order of element needs to be discussed. Finally, the reliability and advantages of the proposed XFEM schemes are illustrated with several 2D and 3D mean compliance minimization examples that are widely used in the recent literature of structural topology optimization. All numerical results demonstrate that the proposed XFEM is a promising structural analysis approach for structural optimization with the level set method.