力学与实践 ›› 2011, Vol. 33 ›› Issue (1): 74-77.DOI: 10.6052/1000-0879-lxysj2010-115

• 教育研究 • 上一篇    下一篇

关于弹性梁的数学模型

刘延柱1,薛纭2   

  1. 1. 上海交通大学工程力学系,200240
    2. 上海交通大学工程力学系 200030
  • 收稿日期:2010-03-22 修回日期:2010-04-16 出版日期:2011-02-15 发布日期:2011-03-03
  • 通讯作者: 刘延柱

On Mathematic Modeling of Elastic Beam

1, 2   

  1. 1. Shanghai Jiao Tong University
    2.
  • Received:2010-03-22 Revised:2010-04-16 Online:2011-02-15 Published:2011-03-03

摘要: 叙述和比较一维弹性体的两种不同建模方法, 即弹性梁的传统建模方法和基于 Kirchhoff-Cosserat模型的建模方法. 应用精确Cosserat模型分析梁的三维运动. 考虑中 心线的拉伸压缩变形、截面的剪切变形、截面转动的惯性和端部载荷影响等因素, 建立精确 的弹性梁动力学方程. 讨论梁的静态和动态平衡稳定性. Kirchhoff杆、铁摩辛柯 梁和欧拉--伯努利梁等为Cosserat模型在各种简化条件下的特例.

关键词: 弹性梁动力学,Kirchhoff动力学比拟,精确Cosserat模型,梁的横向振动

Abstract: The traditional model of beams and the Cosserat model of elastic rods are compared in this paper as two different modeling approaches for one-dimensional elastic body. The three-dimensional motion of a beam is analyzed considering all influence factors, such as the tensile or compressed deformation of the centerline, the shear deformation and the inertia of rotation of the cross section, and the loads on the ends. The stability conditions and free frequency of lateral vibration of a beam are derived on the basis of exact Cosserat model. The Bernoulli beam, Timoshenko beam, and Kirchhoff rod are simplifications of exact Cosserat model under different conditions.