圆截面弹性螺旋杆的稳定性与振动

Stability and vibration of an elastic helical rod with circular cross section

基于Kirchhoff理论讨论圆截面弹性螺旋杆的动力学问题.以杆中心线的Frenet坐标系为参考系,建立用欧拉角描述的弹性杆动力学方程.讨论其在端部轴向力和扭矩作用下保持的无扭转螺旋线平衡状态.在静力学和动力学领域内讨论其平衡稳定性问题.还讨论了弹性杆平衡的Lyapunov稳定性和欧拉稳定性两种不同稳定性概念之间的区别和联系.在一次近似意义下证明了螺旋杆在空间域内的欧拉稳定性条件是时域内Lyapunov稳定性的必要条件.导出了解析形式螺旋杆三维弯曲振动的固有频率,为螺旋线倾角和受扰挠性线波数的函数.

The dynamical behavior of an elastic helical rod with circular cross section are discussed on the basis of Kirchhoff＇ s theory. The dynamical equations of the rod described by the Euler＇ s angles are established in the Frenet coordinates of the centerline. The helical state without twisting of the rod under the action of axial force and torque is discussed. The stability of the helical equilibrium is analyzed in the fields of statics and dynamics respectively. The difference and relationship between Lyapunov＇ s and Euler＇s stability concepts of the rod equilibrium are discussed. We proved in the sense of first approximation that the Euler＇ s stability conditions of the helical rod in the space domain are the necessary conditions of Lyapunov＇ s stability in the time domain. The free frequency of three-dimensional flexural vibration of the helical rod is derived in analytical form as a function of the pitch angle of the helix and the wave number of the perturbed elastica.