Exact solutions for axisymmetric flexural free vibrations of inhomogeneous circular Mindlin plates with variable thickness

Circular plates with radially varying thickness, stiffness, and density are widely used for the structural optimization in engineering. The axisymmetric flexural free vibration of such plates, governed by coupled differential equations with variable coefficients by use of the Mindlin plate theory, is very difficult to be studied analytically. In this paper, a novel analytical method is proposed to reduce such governing equations for circular plates to a pair of uncoupled and easily solvable differential equations of the Sturm-Liouville type. There are two important parameters in the reduced equations. One describes the radial variations of the translational inertia and fiexural rigidity with the consideration of the effect of Poisson＇s ratio. The other reflects the comprehensive effect of the rotatory inertia and shear deformation. The Heun-type equations, recently well-known in physics, are introduced here to solve the flexural free vibration of circular plates analytically, and two basic differential formulae for the local Heun-type functions are discovered for the first time, which will be of great value in enriching the theory of Heun-type differential equations.

Thermal Effect on Vibration of Parallelogram Plate of Bi-Direction Linearly Varying Thickness

In this paper, the effect of thermal gradient on the vibration of parallelogram plate with linearly varying thickness in both direction having clamped boundary conditions on all the four edges is analyzed. Thermal effect on vibration of such plate has been taken as one-dimensional distribution in linear form only. An approximate but quiet convenient frequency equation is derived using Rayleigh-Ritz technique with a two-term deflection function. The frequencies corresponding to the first two modes of vibration of a clamped parallelogram plate have been computed for different values of aspect ratio, thermal gradient, taper constants and skew angle. The results have been presented in tabular forms. The results obtained in this study are reduced to that of unheated parallelogram plates of uniform thickness and have generally been compared with the published one.

NON-LINEAR DYNAMIC BEHAVIOR OF THERMOELASTIC CIRCULAR PLATE WITH VARYING THICKNESS SUBJECTED TO NONCONSERVATIVE LOADING

The non-linear dynamic behaviors of thermoelastic circular plate with varying thickness subjected to radially uniformly distributed follower forces are considered. Two coupled non-linear differential equations of motion for this problem are derived in terms of the transverse deflection and radial displacement component of the mid-plane of the plate. Using the Kantorovich averaging method, the differential equation of mode shape of the plate is derived, and the eigenvalue problem is solved by using shooting method. The eigencurves for frequencies and critical loads of the circular plate with unmovable simply supported edge and clamped edge are obtained. The effects of the variation of thickness and temperature on the frequencies and critical loads of the thermoelastic circular plate subjected to radially uniformly distributed follower forces are then discussed.

Study of Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Parabolically Thickness Variation in Both Directions

The present analysis demonstrates the thermal effect on vibrations of a symmetric, non-homoge- neous trapezoidal plate with parabolically varying thickness in both directions. The variation in Young’s modulus and mass density is the main cause for the occurrence of non-homogeneity in plate’s material. In this consideration, density varies linearly in one direction. The governing differential equations have been derived by Rayleigh-Ritz method in order to attain fundamental frequencies. With C-S-C-S boundary condition, a two term deflection function has been considered. The effect of structural parameters such as taper constants, thermal gradient, aspect ratio and non-homogeneity constant has been investigated for first two modes of vibration. The obtained numerical results have been presented in tabular and graphical form.

Three-Dimensional Vibration of Thick Plate

By the method of initial functions(MIF) and based upon the basic equations of three dimensional theory of elasto dynamics,the governing differential equations of plate with arbitrary thickness are formulated in this paper.The dynamic response of stress and displacement of thick plate subjected to the transverse forces is obtained.It is shown that the vibration characteristics of thick plate consist of three modes: thickness shear mode, symmetric mode and anti symmetric mode.The characteristic equations of simply supported thick plate are derived and the comparison of the free vibration frequencies of moderate thick plate theory and three dimensional elasticty theory with the classical theory is made.

THREE DIMENSIONAL ELASTICITY SOLUTION FOR VIBRATION PROBLEM OF THICK PLATE

In this paper, based upon the basic equations of three dimensional theory of elastodynamics, the governing differential equations of thick plate have been formulated The dynamic response of stress and displacement of thick plate subjected to the transversed forced are obtained. It is shown that the vibrational characters of thick plate consist of three modes: thickness shear mode, symmetric mode and anti-symmetric mode. The characteristic equations;of simply supported thick plate are derived and rile comparison of the free vibration frequencies based on the classic. theory, middle thickness plate theory and three dimensional elasticity theory are given.

SEMI-ANALYTICAL SOLUTION FOR FREE VIBRATION OF THICK FUNCTIONALLY GRADED PLATES RESTED ON ELASTIC FOUNDATION WITH ELASTICALLY RESTRAINED EDGE

This paper deals with free vibration analysis of functionally graded thick circular plates resting on the Pasternak elastic foundation with edges elastically restrained against translation and rotation.Governing equations are obtained based on the first order shear deformation theory（FSDT） with the assumption that the mechanical properties of plate materials vary continuously in the thickness direction.A semi-analytical approach named differential transform method is adopted to transform the differential governing equations into algebraic recurrence equations.And eigenvalue equation for free vibration analysis is solved for arbitrary boundary conditions.Comparison between the obtained results and the results from analytical method confirms an excellent accuracy of the present approach.Afterwards,comprehensive studies on the FG plates rested on elastic foundation are presented.The effects of parameters,such as thickness-to-radius,material distribution,foundation stiffness parameters,different combinations of constraints at edges on the frequency,mode shape and modal stress are also investigated.

一类矩形中厚板模型屈曲与振动问题的辛解析解

从实际力学问题出发抽象出一类矩形中厚板模型,并用辛体系方法进行求解。首先,通过适当的变换将该模型转化为Hamilton系统,得到了Hamilton算子的本征值和本征函数系,并证明了本征函数系的完备性,进而给出对边简支情形的一般解,最后对6种不同边界条件下的Mindlin板和Pasternak型双参数弹性地基的矩形中厚板的振动及屈曲问题进行了数值模拟。数据对比显示了本文模型的正确性和辛方法的有效性。

部分弹性地基Mindlin板自由振动的Hamilton方法

采用Hamilton方法研究Pasternak型部分弹性地基上Mindlin板自由振动问题。引入合适的状态向量,将Mindlin板的控制微分方程转化为相应的Hamilton形式,进一步求解Hamilton算子的本征值和本征函数系,并给出具体的数值算例,数值模拟了部分弹性地基上板的固有频率。

单向变厚度Levy型薄板的自由振动分析

针对单向变厚度Levy型薄板的自由振动问题,基于薄板振动理论,将设定的挠度函数代入关于挠度的变系数四阶偏微分的振动控制方程,把变系数四阶偏微分方程求解挠度的问题转化为第二类Volterra积分方程的求解,并采用二次样条函数近似求解积分方程,建立单向变厚度Levy型薄板自由振动固有频率的求解方法。对3种不同边界条件的Levy型薄板最低固有频率的算例进行验证。研究结果表明:该方法合理可靠、计算简便,满足精度要求;该方法还可进一步推广到求解任意单向变刚度Levy型薄板自由振动的最低固有频率。

线性变厚度矩形薄板自由振动的精确解

基于小挠度薄板理论，采用Ｌｅｒｙ法结合Ｆｒｏｂｅｎｉｕｓ法构造的幂级数解，得到了两对边简支另两对边为ＳＳ、ＣＳ、ＦＦ支承的三种线性变厚度矩形薄板的自振频率随板的边长比及厚度比变化的精确解及其振型函数的解析表达式。

圆板轴对称自由振动的微分容积解法

本文用一种新型的数值方法———微分容积法求解任意边界条件下圆形中厚板的轴对称自由振动问题。通过微分容积法将中厚板轴对称自由振动的控制微分方程和边界条件离散成为一组关于各配点位移的齐次的线性代数方程 ,这是一典型的特征值问题 ,求解该特征值问题便可求得板的自振频率和振型。文中用一些数值算例研究了该方法的收敛性和数值精度 ,展示了该方法的可行性和有效性。

厚板振动的三维弹性力学解

本文以弹性动力学的基本方程为基础，推导出厚板的控制方程·给出了在横向强迫力作用下，厚板的应力、位移的动力响应·得出厚板的振动特征是由对称振动、反对称振动和剪切振动的三种模式组成·最后，将简支厚方板作为实例导出自振频率的特征方程，数值计算结果与经典理论、中厚板理论的结果作了比较·

线性变厚度粘弹性矩形板在随从力作用下的动力稳定性

利用粘弹性微分型本构关系和薄板理论,对线性变厚度粘弹性矩形薄板建立了在切向均布随从力作用下的运动微分方程,采用微分求积法研究了在随从力作用下线性变厚度粘弹性矩形薄板的稳定性问题,具体对对边简支对边固支和三边简支一边固支条件下体变为弹性、畸变服从Kelvin-Voigt模型的变厚度粘弹性矩形板在随从力下的广义特征值问题进行了求解,分析了薄板的长宽比、厚度比及材料的无量纲延滞时间的变化对随从力作用下矩形薄板的失稳形式及相应的临界荷载的影响.

矩形中厚板自由振动问题的哈密顿体系与辛几何解法

以矩形中厚板的胡海昌方程为基础,将中厚板自由振动问题导入哈密顿体系,然后利用辛几何中的分离变量和本征函数展开的方法求出了对边简支板自由振动的精确解.文中采用的辛方法不必事先人为地引入试函数,而是通过完全理性的推导,从而突破了传统半逆解法的限制,使得问题的求解更加合理,易于推广.计算实例证明了本文推导结果的正确性.

厚/薄板振动分析的样条无单元法

建立了厚/薄板振动分析的样条无单元法计算格式,对整个求解区域仅需少量的结点离散,无需单元分划,把挠度和剪应变作为全局的场变量,采用双向三次B样条基函数乘积的线性组合构造场函数,可以充分发挥无单元法及三次B样条基函数的优点。计算结果表明该方法适用于不同厚度,不同边界的板的动力特性分析,无剪切闭锁现象,且精度高,收敛快,未知量少,程序简便。

变厚度圆环板的面内自由振动分析

基于二维线弹性理论,导出厚度沿径向线性变化的变厚度圆环板在面内自由振动的控制微分方程.用微分求积法(DQM)对微分方程及其典型边界条件进行离散,研究变厚度圆环板面内自由振动的无量纲频率特性.数值计算得到不同边界条件下内外半径比、厚度变化参数等因素对无量纲频率的影响.结果表明,圆环板内外边界在夹紧—夹紧和自由—夹紧条件下无量纲频率Ω随厚度变化参数α的增大而增大,在自由—自由和夹紧—自由条件下无量纲频率Ω随厚度变化参数α的增大而减小.

FGM中厚圆板轴对称自由振动的打靶法求解

研究了由陶瓷和金属两种材料组成的功能梯度材料(FGM)中厚圆板的自由振动问题。基于考虑横向剪切变形中厚板的几何方程、物理方程及平衡方程,建立了以中面转角和横向位移为基本未知量的功能梯度中厚圆板轴对称自由振动问题的控制方程;假定功能梯度中厚圆板的材料性质方向按照幂函数连续变化规律;采用打靶法数值求解所得非线性两点边值问题出,获得了多种边界下功能梯度中厚圆板的无量纲自然频率以及振动模态。讨论了材料梯度指数、板的厚度以及边界条件对自然频率的影响。

非均质中厚板的无网格LRPIM动力学分析(英文)

用局部加权残值法建立了非均质中厚板的局部径向点插值离散系统方程,采用无网格局部径向点插值法分析了非均质中厚板的自由振动和强迫振动问题。用径向基函数耦合多项式基函数来近似试函数,用四次样条函数做为加权残值法中的权函数。所构造的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件。该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行。在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化。计算结果表明,利用该方法计算非均质中厚板的自由振动和强迫振动问题可以得到具有较高精度的解。

变厚度环板在面内变温下的固有频率

本文采用薄板线性振动理论分析了厚度沿径向变化和受面内约束的各向同性环形薄板在均匀变温下的自由振动和热弹性稳定性问题。分别针对厚度按指数函数和线性函数变化的两种情况，用有限差分法计算了变厚度环板在内、外周边横向夹紧和简支条件下的线性固有频率和临界温度。由数值结果表明，固有频率的平方与面内变温成线性关系。厚度变化参数、内外半径的比值均对环板的固有频率和临界温度有显著影响。文中给出了丰富的数值结果，可供工程设计参考。