Finite Deflection of Skew Sandwich Plates on Elastic Foundations by the Galerkin Method

ABSTRACT

Application of the Galerkin method to various fluid and structural mechanics problems that are governed by a single linear or nonlinear differential equation is well known [1-5]. Recently, the method has been extended to finite element formulations [6-10], In this paper the suitability of the Galerkin method for solution of large deflection problems of plates is studied. The method is first applied to investigate large deflection behavior of clamped isotropic plates on elastic foundations. After validity of the method is established, it is then extended to analyze problems of large deflection of clamped skew sandwich plates, both with and without elastic foundations. The plates are considered to be subjected to uniformly distributed loads. The governing differential equations for the sandwich plate in terms of displacements in Cartesian coordinates are first established and then transformed into skew coordinates. The nonlinear differential equations of the plates are then transformed into nonlinear algebraic equations, using the Galerkin method. These equations are solved using a Newton-Raphson iterative procedure. The parameters considered herein for large deflection behavior of skew sandwich plates are the aspect ratio of the plate, Poisson's ratio, skew angle, shearing stiffnesses of the core, and foundation moduli. Numerical results are presented for skew sandwich plates for various skew angles and aspect ratios. Simplicity and quick convergence are the advantages of the method, in comparison with other much more laborious numerical methods that require extensive computer facilities.     

基于无网格Galerkin法求解矩形绝缘管道内的磁流体流动

无量纲磁流体流动控制方程中的哈特曼数较大将导致数值计算发散或误差过大。将无网格Galerkin法引入绝缘管道内的稳定磁流体流动计算中,针对磁流体控制方程中大哈特曼数导致计算误差大的情况,对无网格Galerkin法添加了稳定项。计算结果表明,同等条件下,添加了稳定项的无网格Galerkin法总体相对误差远小于标准无网格Galerkin法的结果,且可以计算哈特曼数最大达50000绝缘管道内的磁流体流动。     

Double-mode modeling of chaotic and bifurcation dynamics for a simply supported rectangular plate in large deflection

This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate is first derived and simplified to a set of two ordinary differential equations by the Galerkin method. Several different features including Fourier spectra, state-space plot, Poinca?e map and bifurcation diagram are then numerically computed by using a double-mode approach. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of a rectangular plate possesses many bifurcation points, two different chaotic motions and some jump phenomena under various lateral loading. The results of numerical simulation indicate that the computed bifurcation points can lead to either a transcritical bifurcation or a pitchfork bifurcation for the motion of a large deflection rectangular plate. Meanwhile, the points of pitchfork bifurcation can gradually lead to chaotic motion in some specific loading conditions. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of a large deflection plate.     

A smoothed Hermite radial point interpolation method for thin plate analysis

A smoothed Hermite radial point interpolation method using gradient smoothing operation is formulated for thin plate analysis. The radial basis functions augmented with polynomial basis are used to construct the shape functions that have the important Delta function property. The smoothed Galerkin weakform is adopted to discretize the governing partial differential equations, and a curvature smoothed operation is developed to relax the continuity requirement and achieve accurate bending solutions. The approximation based on both deflection and rotation variables make the proposed method very effective in enforcing the essential boundary conditions. The effects of different numbers of sub-smoothing-domains created based on the triangular background cell are investigated in detail. A number of numerical examples have been studied and the results show that the present method is very stable and accurate even for extremely irregular background cells.     

Postbuckling of rectangular plates under uniaxial compression combined with lateral pressure

Based on the nonlinear large deflection equations of von Kármán plates, the lateral pressure is first converted into an initial deflection by Galerkin method, the postbuckling behavior of simply supported rectangular plates under uniaxial compression combined with lateral pressure is then studied applying perturbation method by taking deflection as perturbation parameter. Two types of in-plane boundary conditions and the effects of initial geometric imperfection are also considered. It is found that the theoretical results are in good accordance with experiments.     

The double mode model of the chaotic motion for a large deflection plate

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Viscoelastic analysis of von Kármán laminated plates under in-plane compression with initial deflection

The creep responses of simply-supported cross-ply and angle-ply viscoelastic laminates, having various magnitudes of imperfections, under in-plane compression are examined. The non-linear strain-displacement relation is based on the von Kármán assumption. The stress function is obtained by solving the compatibility equation by the use of the Laplace transform, and the deflection is calculated from the moment equation by the Galerkin method and a numerical integration scheme. The numerical results of deflection history and edge shortening for the glass/epoxy laminates are presented for illustrating the effect of imperfections and viscoelastic properties on the creep behavior. The solutions based on the quasi-elastic approach are also presented for comparison.     

Non-linear analysis of functionally graded plates under transverse and in-plane loads

Large deflection and postbuckling responses of functionally graded rectangular plates under transverse and in-plane loads are investigated by using a semi-analytical approach. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on the classical plate theory, accounting for the plate-foundation interaction effects by a two-parameter model (Pasternak-type), from which Winkler elastic foundation can be treated as a limiting case. A perturbation technique in conjunction with one-dimensional differential quadrature approximation and Galerkin procedure are employed in the present analysis. The numerical illustrations concern the large deflection and postbuckling behavior of functional graded plates with two pairs of constituent materials. Effects played by volume fraction, the character of boundary conditions, plate aspect ratio, foundation stiffness, initial compressive stress as well as initial transverse pressure are studied.     

轴向扩散下简支饱和多孔弹性梁的大挠度分析

基于多孔介质理论和弹性梁的大挠度理论,并考虑轴向变形,在孔隙流体仅沿轴向扩散的假设下,建立了微观不可压饱和多孔弹性梁大挠度弯曲变形的一维非线性数学模型.在此基础上,忽略饱和多孔弹性梁的轴向应变,并利用Galerkin截断法,研究了两端可渗透的简支饱和多孔弹性梁在突加横向均布载荷作用下的拟静态弯曲,给出了饱和多孔梁弯曲时挠度、弯矩和轴力以及孔隙流体压力等效力偶等沿轴线的分布曲线.揭示了大挠度非线性和小挠度线性模型的结果差异,指出大挠度非线性模型的结果小于相应小挠度线性模型的结果,并且这种差异随着载荷的增大而增大.计算表明:当无量纲载荷参数q>5时,应该采用大挠度非线性数学模型进行研究.     

三维液体非线性晃动动力学特性的数值模拟

主要讨论圆筒形贮腔中三维液体非线性晃动问题,将任意的拉格朗日-欧拉（即ArbitraryLagrangian-Eulerian,简称ALE）运动学描述引入到Navies-Stokes方程中,在时间域上采用一种速度和压力的分步计算格式进行时间离散;在空间域上利用Galerkin加权余量法对系统方程进行数值离散;得到了数值计算粘性不可压液体非线性晃动的ALE分步有限元法的计算格式,推导了三维液体自由液面上结点法向矢量的数值计算方法,模拟了圆筒形贮腔（包括带圆环形隔板的圆筒形贮腔）中三维液体的非线性晃动;并得到了一些重要的非线必不知所云 性,通过数值模拟结果与实验结果的比较,证明实了本文方法的可靠性与有效性。     

AN APPROXIMATION METHOD FOR THE LARGE DEFLECTION OF CANTILEVER BEAMS

研究了悬臂梁自由端受集中力作用时的大挠度变形问题,对大挠度的界定方法做出了一些讨论,并从计算数据分析和理论推导两方面归纳出一种不通过复杂计算就能对大挠度变形进行定量估计的方法. 分析表明,由挠曲线近似微分方程得出的自由端挠度值与梁长度之比值的平方,可以近似表示小挠度法计算挠度值偏离精确挠度值的误差,并由此得出大挠度变形的估计值. 该方法避免了复杂的微分方程求解和数值计算,有一定的工程实际意义.     

轴向移动局部浸液单向板的1:3内共振分析

考虑单向板的轴向速度、轴向张力、流固耦合作用以及阻尼等因素, 基于由 von Kármán薄板大挠度方程得到的轴向移动局部浸液单向板的非线性振动方程, 研究了外激励作用下单向板在1:3内共振情况时的非线性振动特性. 首先利用Galerkin法对非线性振动方程离散化, 然后分别应用数值法和近似解析法对离散后模态方程组进行求解, 获得了系统内共振情况下复杂的幅频特性曲线, 并讨论了周期解的稳定性. 最后研究了1:3内共振系统平均方程组的运动分岔现象.     

载流圆板在磁场中的随机稳定性分析

本文根据大挠度板壳力学基础理论和电磁弹性力学理论,建立了载流圆板的非线性磁弹性随机振动力学模型,采用伽辽金变分法将其变换成非线性常微分动力学方程．通过拟不可积哈密顿系统的平均理论将该方程等价为一个一维伊藤随机微分方程．通过计算该方程的最大Lyapunov 指数判断该系统的局部随机稳定性,并进一步采用基于随机扩散过程的奇异边界理论判断该系统的全局稳定性．最后通过讨论该系统的稳态概率密度函数图的形状变化讨论了该动力系统的随机Hopf分岔的变化规律,并采用数值模拟对理论分析进行了验证．     

The Generalized Nonlinear Galerkin Method Applied to Hydrodynamics of the Open Sea

In this paper we present new numerical algorithms based on a generalized nonlinear Galerkin method in order to solve coastal and oceanic circulation problems. The equations system is based on the primitive equations of the ocean under Boussinesq and hydrostatic approximations. These equations are transformed using, at the same time, the classical σ transformation and an original homogenization of the boundary conditions. We use a well adapted special basis to apply the usual Galerkin method and the nonlinear Galerkin method. This basis is built on a modelization of the energetic transfers through the different scales of flow. Two approaches are proposed to solve the continuity equation: the (nonlinear) Galerkin method and the method of the characteristics. We present the advantages and drawbacks of both methods.     

Large deflection analysis of unsymmetrically laminated composite plates: analytical-numerical type approach

Large deflection analysis of laminated composite plates is considered. The Galerkin method along with Newton-Raphson method is applied to large deflection analysis of laminated composite plates with various edge conditions. The von Kármán plate theory is utilized and the governing differential equations are solved by choosing suitable polynomials as trial functions to approximate the plate displacement functions. The solutions are compared to that of Dynamic Relaxation and finite elements. A very close agreement has been observed with these approximating methods. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems.     

Dynamic buckling of stiffened plates under fluid-solid impact load

A simple solution of the dynamic buckling of stiffened plates under fluid-solid impact loading is presented. Based on large deflection theory, a discretely stiffened plate model has been used. The tangential stresses of stiffeners and in-plane displacement are neglected. Applying the Hamilton‘ s principle, the motion equations of stiffened plates are obtained. The deflection of the plate is taken as Fourier series, and using Galerkin method, the discrete equations can be deduced, which can be solved easily by Runge-Kutta method. The dynamic buckling loads of the stiffened plates are obtained from Budiansky-Roth ( B-R ) curves.     

饱和多孔弹性Timoshenko梁的大挠度分析

基于微观不可压饱和多孔介质理论和弹性梁的大挠度变形假设,考虑梁剪切变形效应,在梁轴线不可伸长和孔隙流体仅沿轴向扩散的限定下,建立了饱和多孔弹性Timoshenko梁大挠度弯曲变形的非线性数学模型.在此基础上,利用Galerkin截断法,研究了两端可渗透简支饱和多孔Timoshenko梁在突加均布横向载荷作用下的拟静态弯曲,给出了饱和多孔 Timoshenko梁弯曲变形时固相挠度、弯矩和孔隙流体压力等效力偶等随时间的响应.比较了饱和多孔Timoshenko梁非线性大挠度和线性小挠度理论以及饱和多孔 Euler-Bernoulli梁非线性大挠度理论的结果,揭示了他们间的差异,指出当无量纲载荷参数q＞l0时,应采用饱和多孔Timoshenko梁或Euler-Bernoulli梁的大挠度数学模型进行分析,特别的,当梁长细比λ＜30时,应采用饱和多孔Timoshenko梁大挠度数学模型进行分析.     

Nonlinear Dynamic Analysis of MEMS Switches by Nonlinear Modal Analysis

A nonlinear modal analysis approach based on the invariant manifoldmethod proposed earlier by Boivin et al. [10] is applied in this paperto perform the dynamic analysis of a micro switch. The micro switch ismodeled as a clamped-clamped microbeam subjected to a transverseelectrostatic force. Two kinds of nonlinearities are encountered in thenonlinear system: geometric nonlinearity of the microbeam associatedwith large deflection, and nonlinear coupling between two energydomains. Using Galerkin method, the nonlinear partial differentialgoverning equation is decoupled into a set of nonlinear ordinarydifferential equations. Based on the invariant manifold method, theassociated nonlinear modal shapes, and modal motion governing equationsare obtained. The equation of motion restricted to these manifolds,which provide the dynamics of the associated normal modes, are solved bythe approach of nonlinear normal forms. Nonlinearities and the pull-inphenomena are examined. The numerical results are compared with thoseobtained from the finite difference method. The estimate for the pull-involtage of the micro device is also presented.     

一种求解大型结构动力响应的快速计算方法

针对大型工程结构动力响应求解效率较低的问题,提出了一种基于减基法的快速求解方法。该方法对动力学方程进行时间域积分构建减基空间,利用Galerkin映射向减基空间进行投影得到减缩方程,利用减缩方程快速求得原系统的逼近解,从而大大提高了动力学方程的求解效率。该方法还考虑了Galerkin映射下的奇异性计算,文中算例验证了该...     

参-强激励联合作用下输流管的分岔和混沌行为研究

研究输送脉动流的两端固定输流管道在其基础简谐运动激励下的分岔和混沌行为,考虑管道变形的几何非线性和管道材料的非线性因素,推导了系统的非线性运动方程,并应用Galerkin方法对其进行了离散化处理。通过采用数值模拟方法,对系统的运动响应进行仿真,重点探讨了流体平均流速、流速脉动振幅以及基础简谐运动激励振幅对系统动态特性的影响。结果表明,系统在不同的参数下会发生围绕不同平衡点的周期和混沌等运动,并在系统中发现了两条通向混沌运动的途径:倍周期分岔和阵发混沌运动。