四边简支矩形薄板的双Hopf分岔分析

基于奇异性理论,研究了主参数共振-1∶3内共振情形下参数激励与外激励联合作用下四边简支矩形薄板的双Hopf分岔问题.考虑弱阻尼和弱激励的情形,得到了四边简支矩形薄板的分岔方程,给出了四边简支矩形薄板在参数平面μ-σ1上的分岔图.对参数激励与外激励联合作用下四边简支矩形薄板的阻尼系数、外激励、参数激励以及调谐参数进行不同的取值,通过数值模拟得到了四边简支矩形薄板平衡解将发生Hopf分岔,并分岔出周期解,薄板系统的非线性振动形式为周期运动.当四边简支矩形薄板的参数满足给定条件时,我们得到薄板的1∶3共振双Hopf分岔.随后,四边简支矩形薄板将会呈现概周期振动.     

均布荷载下四边固支矩形薄板的挠度研究

为求解四边固支矩形薄板在均布荷载作用下的挠度表达式,本文利用叠加法以利维解为基础将复杂问题分解为多个简单问题,然后进行叠加,获得了四边固支矩形薄板在均布荷载作用下的挠度表达式。利用有限元模拟验证表达式的正确性,研究发现有限元结果与公式结果吻合较好,本文利用叠加法求得的挠度表达式具有较高的求解精度,且表达式具有良好的收敛性。基于现有资源,推导出均布荷载下固支矩形薄板的挠度表达式,可供相关人员参考。     

均布载荷下四边固支矩形薄板的挠度

为求解四边固支矩形薄板在均布载荷作用下的挠度表达式,以利维解为基础利用叠加法将复杂问题分解为多个简单问题,然后进行叠加,获得四边固支矩形薄板在均布载荷作用下的挠度表达式.利用有限元模拟验证表达式的正确性,发现有限元结果与公式结果吻合较好.     

四边固支弹性矩形薄板的自由振动

首先弹性矩形薄板的动力学方程表示成为Hamilton正则方程,然后采用辛几何方法对全状态相变量进行分离变量,并利用得到的共扼辛正交归一关系,求出四边固支弹性矩形薄板的固有频率和振型的解析解表达式．由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性矩形薄板的动力学基本方程出发,直接利用数学的方法求出可以满足四边固支边界条件下薄板的固有频率和振型的解析解表达式,使得问题的求解更加理论化和合理化．此外,还给出了计算实例来验证本文所采用的方法以及所推导出公式的正确性．     

参数激励和外激励联合作用下薄板的非线性动力学

研究了在参数激励和外激励联合作用下四边简支矩形薄板的非线性动力学.基于von Karman理论,推导出了在参数激励和外激励联合作用下四边简支矩形薄板的动力学方程.利用Galerkin法对偏微分方程进行三阶离散,得到一个三自由度的常微分方程.考虑1:2:4内共振-主参数共振-1/2亚谐共振的情况,利用多尺度法得到了薄板系统的六维的平均方程.最后,采用数值方法研究了薄板的周期和混沌运动.结果发现外激励对薄板的混沌运动是敏感的.     

薄板的几何非线性求积元分析

针对薄板非线性迭代计算量很大的问题,依据von Kárman薄板非线性理论构造能量泛函,并用数值积分和数值微分进行离散,得到非线性方程组,从而利用求积元法(Quadrature Element Method,QEM)求解薄板的中等挠度的弯曲和非线性屈曲问题,得到可信的结果.算例表明:在处理薄板几何非线性问题上,QEM计算效率很高,应用潜力很大.     

横向磁场中矩形薄板在分布载荷作用下混沌分析

在建立置于横向稳恒电磁场中,同时受横向均布载荷作用四边简支的金属矩形薄板的受力模型的基础上,推导了金属矩型薄板的磁弹性耦合动力学方程,求得了该模型振动系统的异宿轨道参数方程,并根据Melnikov函数方法,推导并求解了振动系统的异宿轨道的Melnikov函数,最后给出了判断该系统发生Smale马蹄变换意义下混沌运动的条件和混沌运动判据.由此可对矩形薄板在机械载荷和电磁载荷共同作用下的分岔和混沌进行分析.本文给出的方法可以推广到其他不同边界条件和不同外载荷条件下弹性薄板的磁弹性振动问题的研究.     

不同位置四点支承矩形薄板的自由振动特性

研究不同位置四点支承条件下矩形薄板的自由振动特性.首先,在板结构模型的不同位置上引入横向约束弹簧,并设定人工弹簧的刚度值以模拟出四点支承的边界条件.然后,基于二维改进傅里叶级数表示结构的位移容许函数,其中改进部分的正弦附加项可解决以往位移函数在边界上可能存在的求导不连续问题.建立矩形板系统能量对应的泛函,令其取驻值建立线性方程组.最后,求解矩阵特征值问题得到点支承矩形板自由振动频率等参数,给出不同位置四点支承条件下矩形薄板的振动特性.所应用二维改进傅里叶级数法中,位移函数基于改进傅里叶级数展开时的附加项能够提高结果的精度和收敛速度.研究结果为不同位置点支承矩形板的自由振动问题提供一定的参考.     

基于板单元形函数的公路桥梁车桥耦合振动分析方法研究

为研究公路多片式梁桥的车桥耦合振动问题,提出一种基于矩形薄板形函数的车桥耦合振动分析方法.该方法以车轮与桥面接触点为界,将车桥耦合系统分为汽车与桥梁2个子系统,分别采用虚功原理与有限元法建立各自的运动方程,并通过车轮与桥面接触处的位移协调条件及车桥相互作用力的平衡关系相耦合,采用矩形薄板形函数实现车桥接触点位移与桥梁节点位移的联系以及车桥相互作用力的分配,通过迭代求解汽车和桥梁的运动方程得到其动力响应.根据分析方法的计算流程,编制了汽车 桥梁耦合系统的动力分析程序,并通过算例分析验证其可行性.研究结果表明,使用基于矩形薄板形函数的公路桥梁车桥耦合振动分析方法得到的车桥动力响应具有较好的精度,该方法具有广泛的适用性,可为多片式梁桥的车桥耦合振动分析提供一种新思路.     

板架结构复杂屈曲问题

利用Patran和MSC Nastran分析板的屈曲问题.明确利用有限元对板屈曲计算的方法,了解板架结构的长宽比和厚度对板架屈曲能力的影响程度.根据矩形板的中性平衡方程推导出板双轴向受压的临界压力公式,并利用有限元方法对其进行校核.     

Buckling of initially stressed mindlin plates using a finite strip method

The buckling of initially stressed Mindlin plates is considered using a thick finite strip method. The method is compared with a wide variety of published results and for both thin and moderately thick plates excellent accuracy is obtained. Some further results are obtained for initially stressed rectangular plates with two opposite edges simply supported and various support conditions on the remaining sides. In general, it is found that for moderately thick plates, Mindlin's plate theory gives lower buckling loads than those obtained using classical thin plate theory.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

Post buckling of a plate with elastic restraints at two unloaded edges

Theoretical analysis is presented for a thin flat rectangular plate with two unloaded edges subject to various support boundary conditions after buckling. The two loaded ends are uniformly compressed and are simply supported. The two opposite unloaded edges are considered either an equal or unequal degree of elastically restrained against rotation and translation.     

Post-buckling behavior of thin steel plates using computational models

Thin plates loaded in plane will buckle at very small loads, and due to unavoidable out-of-plane imperfections, the theoretical buckling load cannot be observed experimentally. If the plate is adequately supported along its boundaries, it will be able to carry a much higher load than the theoretical buckling load.Computational models can be used to study the post buckling behaviour of thin plate structures up to failure. Failure of such structures is usually due to large out-of-plane deflections, yielding, and rupture. Therefore, the computational model should include the effects of geometric and material nonlinearities. In this paper, the nonlinear finite element analysis program NONSAP and ANSR-III were modified and used in the analysis. Since these programs did not include the suitable elements and material properties to conduct the subject study, new elements and new material properties were added to the programs. In particular, a thin shell element was added and the solution routines were modified to improve its accuracy and efficiency.The modified programs were used on a Super Computer to calculate the post buckling strength of stiffened and unstiffened plates subjected to uniaxial compression, and plates subjected to in plane bending or shear. Crippling of plates subjected to in-plane or eccentric edge compressive loads was also examined. The results from the computational models were compared with test results and reasonable agreements were obtained. A computational model was developed for a multi-story thin steel plate shear wall subjected to cyclic loading and the results from the model were compared with experimental results, and again agreement was achieved.     

Biaxial buckling and post-buckling of composite laminated plates on two-parameter elastic foundations

A post-buckling analysis is presented for a simply supported, composite laminated rectangular plate under biaxial compressive loading and resting on a two-parameter (Pasternak-type) elastic foundation. The analysis uses a perturbation technique to determine the interactive buckling loads and post-buckling equilibrium paths. The initial geometrical imperfection of the plates is taken into account. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated rectangular plates. Typical results are presented in dimensionless graphical form.     

Optimal orientation of orthotropic materials for plates designed against buckling

The paper deals with the optimization of anisotropic plates loaded by in-plane forces and designed against buckling. The internal structure of the plate is seen as a twodimensional locally orthotropic solid and the orientations of the axes of orthotropy are taken as the design variables. The problem of optimization consists in determining the best orientation of the axes of orthotropy from the condition of the maximum behaviour of the critical buckling loads. General properties of the problem, such as multiplicity of the critical parameters and splitting of multiple eigenvalues, are studied. Optimization algorithms are developed to improve the anisotropic properties of the plate. The results of numerically finding the optimal orientation of orthotropic properties are compared with conventional layouts for square and rectangular plates loaded by normal in-plane forces.