圆截面弹性细杆的平面振动

基于Kirchhoff理论讨论圆截面弹性细杆的平面振动．以杆中心线的Frenet坐标系为参考系建立动力学方程．杆作平面运动时，其扭转振动与弯曲振动解耦．讨论任意形状杆的扭转振动和轴向受压直杆在无扭转条件下的弯曲振动，证明直杆平衡的静态Lyapunov稳定性与欧拉稳定性条件为动态稳定性的必要条件．考虑轴向力和截面转动惯性效应的影响，导出弯曲振动的固有频率．

PLANAR VIBRATIONS OF A THIN ELASTIC ROD WITH CIRCULAR CROSS SECTION

Based on the Kirchhoff 's theory the planar vibrations of a thin elastic rod with circular cross section are studied. The dynamical equations of the rod are established in the Frenet coordinates of the centerline as the reference frame. In the case of planar motion the torsional vibration is decoupled from the flexural vibration. The torsional vibration of a rod with arbitrary planar shape and the flexural vibration of a straight rod without torsion under axial compression are discussed. It is proved that conditions of Lyapunov's and Euler's stability of equilibrium of a straight rod in static analysis are necessary conditions of its dynamic stability. The influence of the axial force and the inertial effect of the cross section on the natural frequency of flexural vibration is considered.