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力学与实践 ›› 2011, Vol. 33 ›› Issue (1): 60-63.doi: 10.6052/1000-0879-lxysj2010-340

• 应用研究 • 上一篇    下一篇

扇形截面杆扭转问题的差分线法

吴泽艳1,王立峰2,武哲1   

  1. 1. 清华大学航天航空学院
    2. 南京航空航天大学
  • 收稿日期:2010-08-19 修回日期:2010-11-13 出版日期:2011-02-15 发布日期:2011-03-03
  • 通讯作者: 吴泽艳 E-mail:wuzeyan2000@163.com

THE METHOD OF LINES FOR TORSION PROBLEMS OF PRISMATIC BARS WITH SECTOR CROSS-SECTION

2,Wu Zhe2   

  • Received:2010-08-19 Revised:2010-11-13 Online:2011-02-15 Published:2011-03-03

摘要: 导出了扇形截面杆扭转问题偏微分方程的差分线法常微分方程组, 并解析求解了该方程组, 得到了扭转应力函数的半解析解, 计算了扭转应力及扭转刚度. 计算过程中, 用追赶法计算 常微分方程组的特解, 用公式计算三对角矩阵的特征值与特征向量, 利用实对阵矩阵的特征 向量相互正交的特性避免矩阵求逆计算, 利用复化梯形公式计算扭转刚度. 整个求解过程在 角度方向离散微分方程和用复化梯形公式进行面积积分时引入了误差, 其他求解过程是精确 的. 计算结果与已有结果进行了对比, 显示了算法的正确性. 该算法对工程中扇形截面扭 转杆的设计有一定的实用价值.

关键词: 扇形截面杆 扭转 差分线法

Abstract: The method of lines is used to deal with torsion problems of prismatic bars with sector cross-section. Firstly, ordinary differential equations are obtained from partial differential equations of the torsion problem of prismatic bars with sector cross-sections, and are solved analytically. Secondly, the torsion stress function is derived semi-analytically. Finally, the torsion stiffness of the sector cross-section is obtained. The special solution of the ordinary differential equations is obtained by means of the double sweep method, and the eigenvalues and eigenvectors of the tridiagonal matrix are solved, without using the inverse matrix, based on a theorem that eigenvectors of a real symmetric matrix are orthogonal. The torsion stiffness is solved by a compound trapezoid formula. The numerical results are in good agreement with the existing results, which show the correctness of the algorithm. This method is a kind of semi-analytical method, and can be applied in design of prismatic bars with sectorial cross-section in engineering.

Key words: Prismatic Bars with sectorial cross-section