弹性杆的Greenhill公式与Kirchhoff动力学比拟

GREENHILL FORMULA OF AN ELASTIC ROD AND KIRCHHOFF KINETIC ANALOGY

Kirchhoff动力学比拟思想建立了弹性杆静力学与刚体定点转动动力学之间在概念和方法上的对应关系.受拉扭弹性直杆的平衡比拟于Lagrange重陀螺绕铅锤轴的永久转动.根据一次近似理论,考察了两者稳定判据的建立过程,表明其在稳定性上的比拟是Lyapunov意义上的.在此基础上进一步讨论了两端铰支时拉扭弹性直杆的Euler稳定性,并导出亦由Greenhill首先得到的临界载荷计算公式.表明拉扭弹性直杆在两端铰支时的Euler稳定性不同于Lyapunov稳定性,其主要症结在于Euler稳定性中边界条件不受扰动,而Lyapunov稳定性是初值受扰动,两者有区别.

There are corresponding relations of concepts and methods between elastic rod statics and rigid body dynamics about a fixed point according to the thought of Kirchhoff kinetic analogy. A twisted elastic column in equilibrium is corresponding to a Lagrange heavy gyro rotating around a plumb axis. The equilibrium of an elastic rod of tension and twist is corresponding to permanent rotation of Lagrange heave gyro about vertical axis. According to the first approximation theory, the process of building the two stability criteria were examined, which shows that the kinetic analogy between them is in the sense of Lyapunov stability. Based on this, the euler elastic stability of the tension and twist rod with both ends pinned was further discussed and the formula satisfied by critical load was derived by Greenhill firstly. The results show that for the tension and twist rod, the Euler stability is different from Lyapunov stability in the concept. The difference is that the boundary condition is perturbed in Lyapunov stability and not in Euler stability.