扇形截面杆扭转问题的差分线法  

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

作　　者：吴泽艳[1] 王立峰[2] 武哲[1,2] 

机构地区：[1]清华大学航天航空学院,北京100084 [2]北京航空航天大学航空科学与工程学院,北京100191

出　　处：《力学与实践》2011年第1期60-63,38,共5页Mechanics in Engineering

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.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.

关 键 词：扇形截面杆 扭转 差分线法 

分 类 号：O343[理学—固体力学]