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Instead of the biharmonic type equation, a set of new governing equations and solving method for circular sector plate bending
is presented based on the analogy between plate bending and plane elasticity problems. So the Hamiltonian system can also
be applied to plate bending problems by introducing bending moment functions. The new method presents the analytical solutions
for the circular sector plate. The results show that the new method is effective.
Project supported by National Natural Science Foundation (No. 19732020) and the Doctoral Research Foundation of China. 相似文献
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SYMPLECTIC DUALITY SYSTEM ON PLANE MAGNETOELECTROELASTIC SOLIDS 总被引:1,自引:0,他引:1
By means of the generalized variable principle of magnetoelectroelastic solids, the plane magnetoelectroelastic solids problem was derived to Hamiltonian system. In symplectic geometry space, which consists of original variables, displacements, electric potential and magnetic potential, and their duality variables, lengthways stress, electric displacement and magnetic induction, the effective methods of separation of variables and symplectic eigenfunction expansion were applied to solve the problem. Then all the eigen-solutions and the eigen-solutions in Jordan form on eigenvalue zero can be given, and their specific physical significations were shown clearly. At last, the special solutions were presented with uniform loader, constant electric displacement and constant magnetic induction on two sides of the rectangle domain. 相似文献
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基于对偶变量变分原理,选择积分区间两端位移为独立变量,构造了求解完整约束哈密顿动力系统的高阶保辛算法。首先,利用拉格朗日多项式对作用量中的位移、动量及拉格朗日乘子进行近似;然后,对作用量中不包含约束的积分项采用Gauss积分近似,对作用量中包含约束的积分项采用Lobatto积分近似,从而得到近似作用量;最后,在此近似作用量的基础上,利用对偶变量变分原理,将求解完整约束哈密顿动力系统问题转化为一组非线性方程组的求解。算法具有保辛性和高阶收敛性,能够在位移的插值点处高精度地满足完整约束。算法的收敛阶数及数值性质通过数值算例验证。 相似文献
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In this paper, the eigenequation of notch in Reissner plate is derived by the eigenfunction method. Eigenvalues of different notches with different angles are calculated by Muller iteration method.The expression of stress and strain at the tip of notch in Reissner plate is obtained. 相似文献
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According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified way proposed by Luo(1987),some uncon- ventional Hamilton-type variational principles for dynamics of Reissner sandwich plate can be established systematically.The unconventional Hamilton-type variation principle can fully characterize the initial-boundary-value problem of this dynamics.In this pa- per,an important integral relation is given,which can be considered as the generalized principle of virtual work in mechanics.Based on this relation,it is possible not only to obtain the principle of virtual work in dynamics of Reissner sandwich plate,but also to derive systematically the complementary functionals for five-field,two-field and one-field unconventional Hamilton-type variational principles by the generalized Legender trans- formations.Furthermore,with this approach,the intrinsic relationship among the various principles can be explained clearly. 相似文献
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首先,采用特征函数渐近展开法,推导了Reissner板弯曲界面裂纹尖端附近位移场渐近展开的前两阶显式表达式,并利用所获得的位移场渐近表达式构造了一种可用于Reissner板弯曲界面裂纹分析的奇异单元。然后,将该奇异单元与外部的常规有限单元相结合,开展了含界面裂纹Reissner板弯曲断裂问题的数值分析。奇异单元可以较好地描述裂纹尖端附近的内力场与位移场,其优势是它与常规单元进行连接时不需要使用过渡单元,并且可以直接给出应力强度因子等断裂参数的高精度数值结果。最后,通过两个数值算例验证了本文方法的有效性。 相似文献
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I.GeneralAnalyticSolutionforThickRectangularPlates1.1BasicequationsandboundaryconditionsTheoryofthewidelyusedReissuerplatesisquiteaccurateforengineeringcomputationand,asisshowninFig.l,ithasthefollowingassumDtionsfa)AthickrectangularplatewithmaterialpropertiesofelasticmodulusE,Poisson'sratiop,shearmodulusGanddimensionsofainlength,binw.idth.hinheightisconsiderd.anditsstrainsandstressescanbedescribledinthedeflectionofthemid-planeWandrotationsaroundx-axisandJ.-axisAandac,respectively.b)Thepl… 相似文献
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This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method. 相似文献
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The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem. 相似文献
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The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution. 相似文献
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In this paper, problems of bending of a thin plate under the action of in-plane forces are studied by using the method of multiple scales. 相似文献
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四边任意支承条件下弹性矩形薄板弯曲问题的解析解 总被引:1,自引:0,他引:1
利用辛几何法推导出了四边为任意支承条件下矩形薄板弯曲的解析解。在分析过程中首先把矩形薄板弯曲问题表示成Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出四边为任意支承条件下矩形薄板弯曲的解析解。由于在求解过程中并不需要人为的事先选取挠度函数,而是从弹性矩形薄板弯曲的基本方程出发,直接利用数学的方法求出问题的解析解,使得这类问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文方法的正确性。 相似文献
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弹性矩形板问题的Hamilton正则方程 总被引:1,自引:0,他引:1
为了采用辛算法求出弹性矩形板问题的解析解,中直接从弹性矩形板的控制方程出发推导了弹性矩形板,其中包括弹性矩形薄板和厚板问题以及弹性地基上矩形薄板和厚板问题的Hamilton正则方程,为利用辛几何方法求出任意边界条件下这类问题的理论解奠定了基础. 相似文献
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When the Poisson matrix of Poisson system is non-constant, classical symplectic methods, such as symplectic Runge-Kutta method, generating function method, cannot preserve the Poisson structure. The non-constant Poisson structure was transformed into the symplectic structure by the nonlinear transform.Arbitrary order symplectic method was applied to the transformed Poisson system. The Euler equation of the free rigid body problem was transformed into the symplectic structure and computed by the midpoint scheme. Numerical results show the effectiveness of the nonlinear transform. 相似文献
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I.IntroductionThemethodofseparationofvariablesisimportanttosolvethesoluti0n0fprobIem0fmathematicalphysics,butmanyproblen1sofmathematicalphysicscannotseparatet'ariab1es,thereforeitrestrictstheranget0appIicatemethodofseparationofvariable.Inthepaperlll,Zhong… 相似文献
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本文用边界元法研究非均质无限域弹性薄板弯曲问题.在数值实施过程中,对于夹杂和基体分别形成边界积分方程.通过离散边界积分方程,得到相应的方程组,然后结合界面条件,最终获得问题的求解方程组.在界面的相关量求得之后,可以根据需要来求解基体和夹杂中的有关位置的弯矩.数值结果与已有的解做了对比. 相似文献