梁振动问题的Hermite有限元方法.pdf

梁振动问题的 Hermite 有限元方法 摘要 梁振动问题是与我们日常生活密不可分的一类问题，梁结构关系着整个工程的成 败，关系着人民的生命财产安全，关系着国家交通事业与金融经济稳定发展的速度， 因此必须引起我们足够的重视。梁振动方程是在我们所学过的一系列偏微分方程的基 础上延伸出来的一类方程，时间为二阶，空间为四阶，通过对梁振动方程的整体研究， 能够更好地考察对偏微分方程知识的实际应用能力。 在本文中，我主要针对在匀速移动荷载模型下的梁振动问题进行了研究，主要分 为两个部分。第一部分主要研究的是普通的梁的横向振动问题，探索其推导过程以及 在匀速移动荷载模型下的广义解，其中梁振动方程中关于时间的导数为二阶，所以用 到了 Hermite 插值求解基函数即三次元，并利用虚功原理结合 Galerkin 有限元方法对 方程进行变分，再对方程进行半离散化以及全离散化过程，并利用 Matlab 编程进行数 值求解，给出整体以及固定时间点的函数变化图像，将数据结果更直观的呈现出来。 第二部分中引入了加入阻尼的梁振动问题，同样探索其推导过程以及在匀速移动荷载 模型下的广义解，更好地了解在匀速移动荷载下梁的振动规律。 关键词：匀速移动荷载模型，虚功原理，有限元方法，Hermite ，Matlab 1 Hermite finite element method for beam vibration problem Abstract The beam vibration problem is a kind of problem inseparable from our daily life, the safety and stability of the beam is related to the success of the whole project, related to the safety of peoples lives and property, related to the national transportation and financial and economic stability development speed, so we must pay enough attention to it. Beam vibration equation is a kind of equation which is extended on the basis of a series of partial differential equations we have learned. The time is second order and the space is fourth order. In this paper, I mainly focus on the beam vibration problem under the uniform moving load model, which is mainly divided into two parts. The first part mainly studies the common beam transverse vibration problems, explore its derivation process and under the uniform moving load model of generalized solution, of which the results about time for the second order derivative equation, so the Hermite interpolation are used to solve the basis function is three yuan, and using the virtual work principle in combination with Galerkin finite element method (fem) to the variational equation, and then to semi discre