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.     

A DXDR large deflection analysis of uniformly loaded square, circular and elliptical orthotropic plates using non-uniform rectangular finite-differences

A finite-difference analysis of the large deflection response of uniformly loaded square, circular and elliptical clamped and simply-supported orthotropic plates is presented. Several types of non-uniform (graded) mesh are investigated and a mesh suited to the curved boundary of the orthotropic circular and elliptical plate is identified. The DXDR method-a variant of the DR (dynamic relaxation) method-is used to solve the finite-difference forms of the governing orthotropic plate equations. The DXDR method and irregular rectilinear mesh are combined along with the Cartesian coordinates to treat all types of boundaries and to analyze the large deformation of non-isotropic circular/elliptical plates. The results obtained from plate analyses demonstrate the potential of the non-uniform meshes employed and it is shown that they are in good agreement with other results for square, circular and elliptical isotropic and orthotropic clamped and simply-supported plates in both fixed and movable cases subjected to transverse pressure loading.     

Transient bending of a piezoelectric circular plate

Thin, piezoelectric circular plates are frequently used as active components in transducer and smart materials applications. This paper reports on the exact, explicit solution for the transient motion of a piezoelectric circular plate, built-in or simply supported on the edge and electrically grounded over the entire surface. Expressed by elementary Bessel functions and obtained via exact inverse Laplace transforms, the solution enables the efficient calculation of accurate system parameters.     

Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method

The free vibration of a circular plate with multiple perforations is analyzed by using the Rayleigh-Ritz method. Admissible functions are assumed to be separable functions of radial and tangential coordinates. Trigonometric functions are assumed in the circumferential direction. The radial shape functions are the boundary characteristic orthogonal polynomials generated following the Gram-Schmidt recurrence scheme. The assumed functions are used to estimate the kinetic and the potential energies of the plate depending on the number and the position of the perforations. The eigenvalues, representing the dimensionless natural frequencies, are compared with the results obtained using Bessel functions, where the exact solution is available. Moreover, the eigenvectors, which are the unknown coefficients of the Rayleigh-Ritz method, are used to present the mode shapes of the plate. To validate the analytical results of the plates with multiple perforations, experimental investigations are also performed. Two unique case studies that are not addressed in the existing literature are considered. The results of the Rayleigh-Ritz method are found to be in good agreement with those from the experiments. Although the method presented can be employed in the vibration analysis of plates with different boundary conditions and shapes of the perforations, circular perforations that are free on the edges are studied in this paper. The results are presented in terms of dimensionless frequencies and mode shapes.     

Exact solutions for certain slipline fields

The paper deals with explicit solutions for the radii of curvature and the Cartesian coordinates of some important slipline fields. The solutions are expressed in terms of functions involving modified Bessel functions of the first kind. The results are tabulated at 15° angular intervals for fields defined by (a) two orthogonal circular arcs of equal radii, and (b) an orthogonal pair of identical logarithmic spirals. Traction-free boundaries associated with these fields are also discussed and explicit solutions are given. Finally, an exact solution is presented for the slipline field in the neighbourhood of a straight limiting line when the base slipline is a circular arc. A problem of two-dimensional machining is also considered as an example.     

Optimal design of stepped circular plates with allowance for the effect of transverse shear deformation

Presented herein is a canonical exact deflection expression for stepped (or piecewise-constant thickness) circular plates under rotationally symmetric transverse loads. The circular plates may be either simply supported or clamped at the edges. As the plates may be very thick or certain portions of the optimal design may become rather thick, the significant effect of transverse shear deformation on the deflections cannot be ignored. This effect was taken into consideration in accordance to the Mindlin plate theory. Based on the analytical deflection expression, necessary conditions are derived for the optimal values of segmental lengths and thicknesses that minimize the maximum deflection of stepped circular plates of a given volume. These optimality conditions are solved using the Newton method for the optimal segmental lengths and thicknesses. Local minima are observed for this nonlinear problem at hand and they may pose some difficulties in getting the solutions. The shear deformation effect increases the plate deflections, but interestingly it affects the thickness variation marginally.     

Plastic limit analyses of circular plates with respect to unified yield criterion

Plastic limit analysis of a circular plate under uniform transverse loading is presented in terms of an unified yield criterion (UYC). Exact and unified solutions of load-carrying capacities, moment fields and velocity fields for simply supported circular plate, clamped circular plate and annular plate in plastic limit state are derived. Moment fields of the three typical circular plates correspond to different distribution fields on the yield curves. Maximum and minimum plastic limit solutions are deduced for the three types of circular plates by selecting the weighted coefficients with upper and lower bound values in the unified yield criterion. Moment fields and velocity fields with respect to the three special criteria, namely the Tresca criterion, the Mises criterion (close-form solution) and the twin shear stress criterion are illustrated and compared. The paper presents an effective analytical method to compute the exact plastic limit solution for circular plates in terms of a piecewise linear yield criterion.     

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.     

Axisymmetric full-range analysis of transverse pressure-loaded circular plates

Incremental equations governing the axisymmetric full-range response of transverse pressure loaded, clamped and simply supported, circular plates are presented. The derivation and implementation of the full-section and layer constitutive equations is described in detail. The solution of the finite-difference approximations to the governing equations using the Dynamic Relaxation (DR) method is briefly outlined together with several refinements for improving computational efficiency. Both the full-section and layer analyses are used to compute plate deflections observed in several tests on uniformly loaded, clamped and simply supported, circular plates. It is observed that the layer analysis predicts the measured deflections more accurately than the full-section analysis. The analytical-experimental deflection correlation appears to be accurate for slender clamped plates and less accurate for stocky clamped and simply supported plates. It is suggested that various uncertainties about the plate test data may account, in part, for the lack of deflection correlation and that more carefully controlled plate tests should be undertaken in order to confirm this speculation.     

三圆盘非线性扭振系统的内共振

建立了具有平方非线性三圆盘扭振系统的数学模型,应用非线性振动平均法,求得了系统满足双重共振条件的一次近似及对应的稳态解。求得了用椭圆函数表示的系统内共振条件下无阻尼自由振动的解析解,指出系统有能量交换现象,存在周期运动、概周期运动以及调幅调相运动。     

任意位置含振子圆板非轴对称振动的积分方程方法研究

运用特殊函数论和富里叶级数理论,采用由贝塞尔函数组成的平方可积空间上的一类完备正交函数系构造圆板的格林函数;应用积分方程和广义函数论,给出在圆板的任意位置带有弹性振子的耦合系统动态特性的积分方程方法.计算实践表明该方法不仅算法简捷、收敛速度快、计算精度高,而且能从整体上对系统的动态特性加以研究,从而为这类结构的优化设计和安全性评估提供有效的途径.     

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.     

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.     

Limit analysis of circular plates subjected to arbitrary rotational symmetric loadings

The lower bound technique of limit analysis has been implemented in order to calculate the critical load factor for loadings on circular plates. The obtained results are applicable to any rotational symmetric loading on clamped or simply supported plates. Both the square and Tresca yield criteria have been considered in the analysis. It has been shown that the calculated lower bound for the critical load factor is the exact answer for the considered problem. In order to perform the calculations, the loading function is expanded in the form of a power series and the critical load factor calculated by use of a numerical technique. Finally, variation of this factor with the loading function has been illustrated.     

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.     

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.     

Pure global buckling and vibration in laminates with arbitrary shape cut off regions

The pure global buckling and vibration of four sides simply supported as well as clamped anisotropic laminates having an arbitrary shape cut off region that is symmetric with respect to mid-plane have been studied by treating the remaining cut off regions as uniform plates with reduced stiffness. The reduced variation of stiffness of the plate is represented by Fourier series. Computational solutions of the energy principle for the Ritz method in a plate having arbitrary shape of typical cut off regions under biaxial compressive loads are obtained. Some numerical results for the pure global buckling load prediction due to its reduced flexural stiffness for the circular cut off regions and elliptical cut off regions are presented. We find the normalized pure global buckling load ratio decreases as the cut off regions size increases, and the non-dimensional fundamental frequency value decreases as the cut off regions size increases.     

Analytical and experimental studies for deformation of circular plates subjected to blast loading

In this paper, experimental responses of the clamped mild steel, copper, and aluminum circular plates subjected to blast loading are investigated. Extensive experimental results concerning variations of central deflection are presented. Approximate energy analysis is used for the analytical modeling of the mid-point deflection thickness ratio of the circular plates. The essence of the model is to describe the deformation profile, which depends on the distribution of impulsive load. By using the present modeling, one can show how the midpoint deflection varies with the variations of the important parameters. The results obtained from the present model show very good agreement with the experimental data.     

Natural frequencies of shear deformable rhombic plates with clamped and simply supported edges

The natural vibrations of thick and thin rhombic plates with clamped and simply supported edges are analyzed, using assemblages of nine-node Lagrangian isoparametric quadrilateral C⁰ continuous finite elements based on a higher-order shear deformable thick plate theory. Here, additional nodal displacement degrees of freedom are derived by retaining higher-order powers of the thickness coordinate in the in-plane displacement fields, which in turn allows for the proper representation of the transverse shear strains of thick plates. Essential rotary inertia terms are derived and included in the present analysis. Nondimensional frequencies are calculated for thick and thin rhombic plates having various combinations of clamped and simply supported edge conditions, and skew angles. The efficacy of using higher-order shear deformable plate finite elements for predicting the in-plane vibration modes of rhombic plates is found to increase as the span-to-thickness ratio decreases and the skew angle increases. The present work shows that higher-order shear deformable finite elements essentially eliminate the transverse shear over-correction of thick rhombic plate frequencies that is produced when finite elements based on the widely used first-order Reissner-Mindlin plate theory are utilized.     

Axisymmetric and asymmetric postbuckling response and energy release rate of delaminated circular film

A delaminated circular film is treated as a thin circular plate clamped to the substrate at the boundary of the delaminated zone. The axisymmetric postbuckling response for the large deflections of the film under the action of in-plane compression has been investigated by using a power series method to solve the Karman nonlinear equations for the axisymmetric deformations of circular plates. The problem of the asymmetric bifurcation-buckling of the film from the axisymmetric large-deflection equilibrium state has been investigated by using Fourier series and power series simultaneously to solve the Karman nonlinear equations for the asymmetric deformations of circular plates.The energy release rate related to the two stages of the buckling deformation has been calculated, and it is found that the energy release rate corresponding to the stage of the asymmetric postbuckling deformation is much higher than that of the axisymmetric postbuckling stage.