松弛状态非圆截面弹性螺旋细杆的稳定性

STABILITY OF A THIN ELASTIC HELICAL ROD WITH NONCIRCULAR CROSS SECTION IN RELAXED STATE

研究松弛状态下的非圆截面弹性螺旋细杆,即带有原始曲率和挠率的非圆截面弹性杆的平衡稳定性问题.基于Kirchhoff动力学比拟,建立用欧拉角表达的弹性杆动力学方程.忽略线加速度引起的微小惯性力,仅考虑截面转动的动力学效应,使欧拉方程封闭.证明松弛状态下的非圆截面螺旋杆无论在空间域或时域均满足一次近似意义下的Lyapunov稳定性条件.从而为螺旋形态弹性细杆存在于自然界中的广泛性和稳定性作出理论解释.提示负泊松比材料的螺旋杆可能不稳定.

This paper discussed the stability of a thin elastic helical rod with noncircular cross .section in relaxed state, i.e., the stability of a rod with intrinsic curvature and twisting. Based on the Kirchhoff＇ s kinetic analogy, the dynamical equations of the elastic rod were expressed by the Euler＇s angles. Neglecting the small force of inertia caused by the linear acceleration, only the inertial effect of the rotation of the cross section was considered, which made the Euler＇s equations closed. We proved that the Lyapunov＇ s stability condition in first approximation was satisfied for the helical rod in relaxed state in the spatial domain, as well as in the time domain. Therefore the extensive and stable existence of a thin elastic rod with helical configuratiori in the nature can be explained theoretically. It was alto noticed that a helical rod with negative Poisson ratio can be unstable.