非圆截面弹性细杆的螺旋线平衡及稳定性

本文研究端部受力和力矩作用，且存在初曲率和初扭率的非圆截面弹性细杆的螺旋线平衡及其稳定性。描述弹性细杆平衡状态的Kirchhoff方程存在与杆的螺旋线平衡状态相对应的特解。直杆和圆环杆为螺旋线状态的两种特例。文中分析了螺旋线的几何特性与作用力和力矩之间的相互关系，并导出螺旋线平衡的一次近似解析形式稳定性判据。分析表明，松弛状态下弹性杆可处于螺旋线状态，直杆只有在轴向压力的作用下才能保持螺旋线平衡。无初曲率和初扭率弹性杆的螺旋线平衡稳定性必要条件是杆截面绕副法线轴的抗弯刚度大于或等于绕法线轴的抗弯刚度。此条件也适用于带初扭率的圆环杆及更普遍情形。无初曲率和初扭率的圆截面杆的螺旋线平衡恒稳定。

Helical Equilibrium and Stability of a Thin Elastic Rod with Noncircular Cross-Section

The helical equilibrium and its stability of a thin elastic rod with noncircular cross section and intrinsic curvature and twisting under application of force and torque were discussed. The Kirchhoff equations describing the equilibrium of elastic thin rod has a particular solution, corresponding to the helical e-quilibrium. The straight rod and the planar ring are two special cases of helix. The relationship between geometric behavior of the helix and the external force and torque was analyzed, and a stability criterion of helical equilibrium in analytical form was derived in first approximation. It is shown that the relaxed state of a rod can be a helical state, and the helical equilibrium of a straight rod can be maintained only under axial pressure forces. The necessary stable condition of helical equilibrium of a rod without intrinsic curvature and twisting is that the bending rigidity of the rod about the binormal axis is larger than the bending rigidity about the normal axis. The same condition applies for a elastic ring with intrinsic twisting and more general cases too. The helical equilibrium of a rod with circular section but without intrinsic curvature and twisting is always stable.