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.     

哈密顿体系下矩形薄板自由振动的一般解

根据弹性薄板自由振动问题的基本方程，把问题引入到哈密顿对偶体系中．x方向模拟为时间，选取弯矩，等效剪力，转角和挠度为对偶向量，得到了在不同边界条件时关于x轴对称和反对称时的解析解．算例研究了四边固支薄板的自由振动情形，从而推广了哈密顿体系的应用范围，验证了哈密顿体系求解方法在自由振动问题中的有效性．     

矩形正交各向异性薄板弯曲受迫振动问题的分析解

通过恰当的辛内积定义,首先将矩形正交各向异性薄板弯曲受迫振动问题导入到辛对偶体系,并应用分离变量和辛本征展开的有效数学物理方法给出其受迫振动稳态解的一个解析求解方法.然后,具体讨论了对边简支和对边固支两种典型边界条件的正交各向异性薄板弯曲受迫振动问题的辛本征问题,并给出了对应的辛本征值超越方程和辛本征向量的解析表达式....     

薄壁圆柱壳的高阶模态振动特性研究

采用解析法研究了不同边界条件下薄壁圆柱壳的高阶模态振动特性.首先基于Love壳体理论,在简支-简支、固支-固支和固支-自由三种边界条件下,通过伽辽金法建立了动力学模型,对模态特性进行求解,得到了高阶固有频率和三维模态振型,并通过文献和有限元法进行了比较.算例结果表明,两端简支边界条件下采用解析法得到的固有频率误差值不超过2%,当周向波数较小时固有频率先减小后增加,在高阶时的固有频率逐渐升高,当轴向半波数增加时固有频率明显增大,通过解析法、文献和有限元法得到的三维模态振型相吻合.     

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.     

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.     

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.     

双参数弹性地基上板的自由振动

建立了双参数弹性地基上的正交异性矩形薄板自由振动位移函数微分方程,并得到其一般解．这可用以精确地求解板在任意边界条件下的自由振动问题．以四边固定的正方形板为例进行了分析,计算过程简单,便于实际应用．亦适用于求解单参数弹性地基和各向同性板情形。     

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．     

辛几何形态下不同边界条件的薄板解析解

利用平面弹性与板弯曲的相似性理论,用直接法研究辛几何形态下的薄板弯曲问题。当薄板对边边界条件形式不同时,将其进行降阶形成对偶方程组,再利用分离变量法把问题转化为本征值问题求解。通过本征函数、辛正交关系、展开求解等手段得到了薄板的解析解。算例表明辛求解的有效性与快速收敛性。     

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．     

Linear and non-linear deflection analysis of thick rectangular plates—II. Numerical applications

Variational methods are widely used for the solution of complex differential equations in mechanics for which exact solutions are not possible. The finite difference method, although well known as an efficient numerical method, was applied in the past only for the analysis of linear and non-linear thin plates. In this paper the suitability of the method for the analysis of non-linear deflection of thick plates is studied for the first time. While there are major differences between small deflection and large deflection plate theories, the former can be treated as a particular case of the latter, when the centre deflection of the plate is less than or equal to 0.2–0.25 of the thickness of the plate. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. In this analysis thin plates are treated as a particular case of the corresponding thick plate when the boundary conditions of the plates are taken into account. The method is first applied to investigate the deflection behaviour of clamped and simply supported square isotropic thick plates. After the validity of the method is established, it is then extended to the solution of rectangular thick plates of various aspect ratios and thicknesses. Generally, beginning with the use of a limited number of mesh sizes for a given plate aspect ratio and boundary conditions, a general solution of the problem including the investigation of accuracy and convergence was extended to rectangular thick plates by providing more detailed functions satisfying the rectangular mesh sizes generated automatically by the program. Whenever possible results obtained by the present method are compared with existing solutions in the technical literature obtained by much more laborious methods and close agreements are found. The significant number of results presented here are not currently available in the technical literature. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. The subroutine SOLINV using the change of variable technique illustrated elsewhere takes care of the solution of the general system. Simplicity in formulation and quick convergence are the obvious advantages of the finite difference formulation developed to compute small and large deflection analysis of thick plates in comparison with other numerical methods requiring extensive computer facilities.     

Application of the extended Kantorovich method to the bending of variable thickness plates

The governing equations of the classical plate theory for a uniform or a unidirectional variable thickness rectangular plate under transverse applied loading are solved by means of the extended Kantorovich method. The plate may be either simply supported or clamped along the edges. The solution procedure is iterative and must be carried out numerically. This necessitates the calculation of the two missing pieces of boundary data along the edges of the plate. The missing boundary data are determined utilizing the method of adjoints of the shooting method. The numerical values of the deflection and bending moments for uniform and variable thickness plates are compared with those from the exact solutions and finite element analysis, respectively.     

Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: the effects of large vibration amplitudes on the fundamental mode shape

The geometrically non-linear free vibrations of thin isotropic and laminated rectangular composite plates with fully clamped edges have been successfully investigated in previous series of works using a theoretical model based on Hamilton’s principle and spectral analysis. The objective of this work is the extension of the above model to the case of clamped clamped simply supported simply supported rectangular plates, denoted by CCSSSSRP, in order to determine their fundamental non-linear mode shape, and associated amplitude-dependent resonant frequencies, and flexural stress distribution. Numerical data are given for both linear and non-linear analysis, for various plate aspect ratios and vibration amplitudes. Good agreement was found with previous published results.     

Nonlinear dynamics of angle-ply composite laminated thin plate with third-order shear deformation

An asymptotic perturbation method is presented based on the Fourier expansion and temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported angle-ply composite laminated rectangular thin plate with parametric and external excitations.According to the Reddy's third-order plate theory,the governing equations of motion for the angle-ply composite laminated rectangular thin plate are derived by using the Hamilton's principle.Then,the Galerkin procedure is applied to...     

A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges

On the basis of three-dimensional elasticity, this paper presents a free vibration analysis of cross-ply laminated rectangular plates with clamped boundaries. The analysis is based on a recursive solution suitable for three-dimensional vibration analysis of simply supported plates. Clamped boundary conditions are imposed by suppressing the edge displacements of a number of planes which are parallel to the mid-plane. This is achieved by coupling a number of different vibration modes of the corresponding simply supported plate using a Lagrange multiplier method. A satisfactory solution can be obtained by choosing suitably larger numbers of the coupled vibration modes and the fixed planes across the thickness of the plate. Numerical results are presented to show the convergence of the solution. Results are also obtained for either isotropic or cross-ply laminated plates having different combinations of simply supported and clamped boundaries.