弹性细杆螺旋线平衡的动态稳定性

Dynamical Stability of Helical Equilibrium of a Thin Elastic Rod

本文从动力学观点讨论具有初扭率的非圆截面弹性细杆的螺旋线平衡稳定性。弹性杆平衡的动态稳定性建立在以弧坐标s 和时间坐标t 为双自变量的离散系统的Lyapunov稳定性概念基础上。对于两端约束状况固定不变的弹性杆,若静态稳定性条件已满足,其与弧坐标对应的本征值可根据端部约束条件确定。则螺旋线平衡的动态稳定性由时间域的本征值判断。在缓慢受扰运动条件下,引入尺度缩小的时间变量T=εt ,可将动力学过程视为对平衡状态的摄动。证明在ε2 计算精度范围内,当螺旋线平衡的一次近似静态稳定性条件得到满足时,考虑动力学因素的稳定性条件必也同时满足。

The problem on stability of helical equilibrium of a thin elastic rod with noncirular cross-section and intrinsic twisting was discussed in view of dynamics. The dynamical stability analysis was based on the concepts of Lyapunov stability for a discrete dynamical system with arc-coordinate s and time t as dual arguments. In the case of the rod with fixed end constraints, and the conditions of static stability are satisfied, the eigenvalues corresponding to the arc-coordinate can be determined by the boundary conditions. Then the dynamic stability of the helical equilibrium depends on the eigenvalues in the time domain. Under the condition of slow perturbed motion, the dynamical process of the rod can be regarded as a perturbation of equilibrium state in statics. Introducing a scale-reduced time variable T=εt, the dynamical equations of the rod and its perturbed equations of helical equilibrium were established on the basis of Kirchhoffs theory. It is proved that in the sense of first approximation with precision of ε 2 when the conditions of stability of helical equilibrium of the rod are satisfied within the scope of statics, the stability conditions with consideration of dynamical factors are satisfied in the same time.