轴向受压螺旋杆的平衡稳定性

STABILITY OF EQUILIBRIUM OF A HELICAL ROD UNDER AXIAL COMPRESSION

在K irchhoff动力学比拟基础上讨论端部受轴向压力作用的圆截面弹性细杆的螺旋线平衡稳定性问题.弹性杆的平衡状态由Eu ler角描述的弹性杆平衡方程的特解确定.从Lyapunov或Eu ler的不同稳定性概念出发,对弹性杆的平衡稳定性的判断可得出不同的结果.根据一次近似扰动方程判断,弹性杆的螺旋线状态和圆环状态恒满足Lyapunov稳定性条件.但螺旋杆在轴向压力到达临界值时,圆环杆在扭转数到达临界值时将产生屈曲而丧失Eu ler稳定性.导出临界载荷和临界扭转数的计算公式.螺旋杆的临界载荷取决于螺旋线的高度和螺旋角.螺旋角趋近于π/2时螺旋杆转化为带扭率的直杆,其临界载荷的极限值与压杆的Eu ler载荷一致.文中对两类不同稳定性概念的区别和联系作出解释.

The stability of helical equilibrium of a thin elastic rod with circular cross section under the axial compression is discussed on the basis of the Kirchhoff＇s kinetic analogy. The equilibrium states of the rod are determined by the special solutions of equilibrium equations described by Euler＇s angles. Different conclusions of stability analysis are obtained according to different stability conceptions of Lyapunov or Euler. It is shown that based on the perturbed equations of first approximation, the Lyapunov＇ s stability conditions are always satisfied for the helical and annular equilibrium states of the rod. In the contrary, the rod buckles and loses the Euler＇ s stability when the axial force or twist number reaches a critical value. The formulas of critical values of axial force and twist number are derived, where the critical load of a helical rod depends on the height and pitch angle of the helix. The helical rod transfers to a straight rod with torsion when the pitch angle tends to π/2. The limit of critical load of the straightened helical rod is consistent with the Euler＇ s load of a compressed straight rod. The distinction and relation of two kinds of stability conceptions are explained.