Kirchhoff弹性直杆在力螺旋作用下的稳定性

研究受力螺旋作用的圆截面Kirchhoff弹性直杆在各种边界条件下的稳定性问题. 用直角坐标和Cardano角表示截面的形心位置和姿态. 由Kirchhoff方程得到弹性细杆的直线平衡特解，导出线性化扰动方程及其通解. 根据边界条件确定积分常数的非零解存在条件，讨论了各种边界条件，如两端铰支、两端固定、一端铰支一端固定以及一端固定一端自由的弹性细杆直线平衡状态的稳定性，导出了临界载荷的表达式，绘制了稳定域，将Greenhill公式推广到其他边界条件，并且使压杆的Euler 公式成为其特例. 关键词： Kirchhoff弹性杆 稳定性 力螺旋 Greenhill公式

Stability of a straight Kirchhoff elastic rod under the force screws

Stability of a straight Kirchhoff elastic rod with circular cross section acted by a pair of force screws is studied. Cartesian coordinate and Cardan angle are used to express the position and attitude of a cross section of the rod. Special solution which is a straight equilibrium state of the rod is derived from Kirchhoff equation of the rod, and the linear perturbation equation on this special solution is further solved. The stability of the solution for the straight equilibrium state of the rod is discussed according to the existence of non-zero solution of integration constants at various kinds of boundary conditions of the rod, such as that with two joints ends, two fixed ends, a fixed end and a free end, or a joint end and a fixed end. The critical loads are deduced and the stable ranges are plotted. Greehill formula is extended to other cases, and Euler formula for compression rod becomes its special case.