Asymmetric vibration and dynamic stability of bimodulus thick annular plates

The free vibration and dynamic stability problems of asymmetric bimodulus thick annular plates are studied. The annular element with Lagrangian polynomials and trigonometric functions as shape function is developed. The element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. The dynamic stability of an annular plate subjected to a combination of a pure dynamic bending and a uniform dynamic extensional stress in the plane of the plate is investigated. The non-axisymmetric modes are shown to have significant effects in the annular bimodulus plates.     

Axisymmetric vibration of bimodulus thick circular and annular plates

The fundamental natural frequencies of axisymmetric circular and annular plates subjected to a combination of a pure bending stress and extensional stress in the plane of the plate are investigated. The thick plate ring element model which includes the effect of transverse shear deformation is created for axisymmetric free vibration problems. The obtained results of non-dimensional natural frequency coefficient compared with the closed form solutions for ordinary plates are shown to be very good. The effects of various parameters on the natural frequency and neutral surface locations are studied. The bimodulus properties are shown to have significant influence on the natural frequency.     

轴向运动薄板非线性振动及其稳定性研究

应用增量谐波平衡法(IHB法)研究轴向运动薄板横向非线性振动特性及其稳定性.通过Hamilton原理推导出了非惯性参考系下四边简支轴向运动薄板的横向振动微分方程,然后利用Galerkin方法离散运动方程.对离散后的非线性方程组应用IHB法进行非线性振动分析,研究了在固有频率之比ω20/ω10接近于3:1情况下,外激励频率ω在ω10附近的具有内部共振的基谐波响应.最后用多元Floquet理论分析了系统周期解的稳定性,其中采用Hsu方法来计算转移矩阵.通过对具体例子的数值计算,分别得到了自由振动和不同外激励下的频幅相应曲线,通过对比运动梁模型和运动薄板模型的计算结果,分析了各种模型的适用范围.     

Flexural vibration of shear deformable circular and annular plates on ring supports

This paper presents an analysis of free flexural vibration of circular and annular Mindlin plates lying on multiple internal concentric ring supports. The Rayleigh-Ritz method with an admissible displacement function expressed in terms of a set of simple polynomials is used to obtain the governing eigenvalue equation. The natural frequencies of circular and annular Mindlin plates with one or two ring supports having inner and outer peripheries with different combinations of free, simply supported and clamped conditions are determined. A comprehensive set of vibration results is presented and, where possible, these have been verified with existing solutions published in the open literature. No previous theoretical results for thick annular plates supported with internal rings are known to exist. Thus the study should be useful and valuable to engineers and researchers.     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

Asymmetric vibration analysis of thick composite circular cylindrical shells with variable thickness

Thick isotropic and composite shells of revolution are analyzed for vibration characteristics. The theory used is that proposed by Naghdi, which includes the thickness normal strain and shear deformation. A curved-semianalytical finite element is used to solve the problem. Parametric studies are conducted to study the effect of various parameters on the vibration behavior of shells. The effect of thickness variation along the axial direction with a constraint on mass is also studied.     

Nonlinear axisymmetric vibration of orthotropic thin circular plates with elastically restrained edges

This paper deals with the large-amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates of varying thicknesses with edge elastically restrained against rotation and in-plane displacement. Geometric nonlinearity due to moderately large deflections is included. Linear, parabolic and cubic variations of thickness are considered. Harmonic vibrations are assumed and time is eliminated from the von Kármán-type governing equations by the Kantorovich averaging method. The orthogonal point collocation method is used for spatial discretisation. Results are presented for the linear frequency of first axisymmetric mode and for the amplitude-period response. The effect of taper ratio, orthotropic parameter and rotational and in-plane stiffness of the support on the nonlinear vibration behaviour is investigated.     

A study of various effects on the stability of circular plates

A study of various effects such as geometry and material property variation, elastic restraints and elastic foundation, and shear deformation on the buckling characteristics of circular plates is presented in this paper. A finite element formulation is employed for the purpose and numerical results, along with a discussion on each of the above effects, are presented. The results are expected to be useful for design engineers.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

On the vibration of annular, circular and sectorial plates with cut-outs or on partial supports

Natural frequency parameters are given for the free vibration of annular, circular and sectorial plates which may be supported along only parts of their boundaries or may have sectorial cut-outs. The results are obtained by using a Ritz solution previously presented for the study of the vibration of cracked circular or annular plates and which makes use of a minimum number of sectorial elements joined together by means of artificial springs, the stiffnesses of which are permitted to become very high, thus satisfying the required continuity conditions.     

Thermal post-buckling of circular plates

The thermal post-buckling behaviour of isotropic circular plates has been studied in this paper through a simple finite element formulation. The accuracy of the solution by finite element method is established through a solution by Rayleigh-Ritz method for simply-supported circular plates. Effect of non-linearity on the load parameter is found to be much higher in the case of thermal loading than in the case of mechanical loading.     

Elastic buckling of tapered circular plates

This paper is concerned with the elastic buckling of tapered circular plates. The study is prompted by the fact that results hitherto available are restricted to a narrow range of taper parameters and are somewhat different from each other. For the buckling analysis, a simple and yet accurate numerical method is presented. It is based on the shooting method and the Rayleigh-Ritz approach. Comprehensive generic buckling results of circular plates with linearly and parabolically varying thicknesses are generated. Comparison studies of the buckling results showed that some of the existing results were erroneous. Optimal values of taper parameters for such tapered plates are also given.     

Asymmetric buckling of bimodulus thick annular plates

An annular element with Lagrangian polynomials and trigonometric functions as shape functions is developed for asymmetric finite element stability analysis. The annular element is based on the Mindlin plate theory so that the effect of transverse shear deformation is included. Using the asymmetric finite element model, the asymmetric static buckling of bimodulus thick annular plates subjected to a combination of a pure bending stress and compressive normal stress is investigated. The obtained results of non-dimensional critical buckling coefficients are shown to be very accurate when compared with the exact solutions. The effects of various parameters on the buckling coefficients are studied. The bimodulus properties are shown to have significant influences on the buckling coefficients.     

Optimal shakedown loading for circular plates

Optimization of shakedown loading under constrained residual displacement is considered for elastic and perfectly plastic circular plates. The load variation bounds, which satisfy the optimality criterion in concert with plate-strength and stiffness requirements, are identified. The actual strain fields of the plate depend on the loading history. Thus, the load optimization problem at shakedown is stated as a pair of problems that are executed in parallel: the main load optimization and the verification of the prescribed magnitudes of the bounds on the residual deflections. The problem must be solved by iteration. The Rozen projected gradient method is applied. A mechanical interpretation of the Rozen optimality criterion is given, which permits the simplification of the mathematical model for load optimization and the formulation of the solution algorithm. Numerical examples include circular annular plates with and without a rigid inclusion. The results are valid under the assumption of small displacements.     

Axisymmetric buckling of bimodulus thick circular plates

The static buckling of bimodulus thick circular and annular plates subjected to a combination of a pure bending stress and compressive stress is investigated. The thick finite element model, which includes the effect of transverse shear deformation, are created for axisymmetric buckling problems. The obtained results of buckling coefficient are compared with the exact solutions for ordinary thin plates. The accuracy of the finite element solutions are shown to be very good. The effects of various parameters on the buckling coefficients and neutral surfce locations are studied. The bimodulus properties are shown to have significant influences on the buckling coefficient.     

悬臂板条结构动力学与振动控制分析

基于Lagrange-Germain弹性薄板理论,采用Hamilton列式求解方法,研究了悬臂板动力学与振动控制问题．确定了平板中纵横振动模式存在的色散关系,给出了问题的解析解．基于乎板振动的构造解,对板条结构的振动实施了主动控制．本文还做了数值仿真,并对结果进行了分析讨论．     

Nonlinear transient response of isotropic circular plates

The present work investigates the nonlinear transient response for stresses and deflection of thin linear elastic circular plates subjected to different time-dependent loads. Space-wise discretisation has been done using collocation method with the zeros of a Chebyshev polynomial as collocation points. Newmark-β scheme has been used for time-marching. Time and space-wise convergence studies have been carried out. Both the linear and nonlinear transient response of clamped and simply supported circular plates under six different sets of transient loads have been obtained. Several new useful results are reported in the present study.     

Bending of circular plates with variable profile

The technique of Dynamic Relaxation is invoked in analysing the bending of circular plates with variable profile and the results are compared with available exact solutions. The excellent correlation suggests that the technique can serve as an alternative for computing the magnitudes and distributions of moments and deflections in circular plates of anisotropic material with variable profile.     

Free vibration analysis of circular cylindrical shells

Determination of the vibrational characteristics of circular cylindrical shells often requires significant computational effort. This paper presents the results of a comprehensive, computer based, numerical investigation of the free vibration of circular cylindrical shells. An analytical procedure which accurately predicts the natural frequencies and radial mode shapes (corresponding to axial wave number and circumferential wave number both equal to one) for a wide range of circular cylindrical shells is developed. The procedure is applicable to shells either with or without a top closure. Several numerical examples are presented which illustrate application of the procedure and verify its accuracy.     

Orthogonal polynomials and sub-sectioning method for vibration of plates

Flexural vibrations of plates with discontinuities in the form of cracks and cutouts are investigated using a computationally efficient and accurate numerical scheme; the scheme combines the flexibility of dividing the problem domain into appropriate segments and the high accuracy resulting from the use of orthogonal polynomial functions generated using the Gram-Schmidt process. The frequency parameters obtained for standard plate problems using the proposed method compare very favourably with existing numerical and analytical results. The scheme is then extended to new cases involving plates with cutouts and cracks subjected to various combinations of inner and outer edge conditions.