Kirchhoff弹性杆分析动力学的准坐标表达

作为DNA的力学模型,依据Kirchhoff动力学比拟思想建立的弹性细杆的分析力学方法已从静力学深入到动力学。由于静力学平衡微分方程与刚体动力学相当,因此,弹性细杆动力学的分析力学方程必是以弧坐标和时间为双自变量的偏微分方程。以横截面的形心速度以及弯扭度和角速度沿主轴的分量为准速度,定义了准坐标,导出了准坐标的微分和变分运算的交换关系。从Hamilton原理出发,利用准坐标的微分和变分运算的交换关系,导出了Kirchhoff弹性杆动力学准坐标下的Boltzmann-Hamel方程,并由此导出Lanrange方程。指出了Boltzmann-Hamel方程显式即为弹性杆动力学的Kirchhoff方程。定义关于弧坐标和时间的正则变量和Hamilton函数,导出Boltzmann-Hamel方程的正则形式。本文结果是以弹性杆静力学和刚性杆动力学为其特例。作为例子,建立了垂挂的在重力作用下作平面运动的弹性细杆的动力学微分方程以说明本文方法的应用。

Analytical Dynamics of a Kirchhoff Elastic Rod Expressed in Quasi-Coordinates

Analytical mechanics method of a super thin elastic rod statics as a DNA model based on the theory of Kirchhoff's kinetic analogy was fully studied. Since the focus study of the rod mechanics has changed from the statics to dynamics, developing analytical mechanics method of a super thin elastic rod dynamies in which the independent variables are arclength of central line of the rod and the time it is necessary. The velocity of the centre, curvature twisting vector and angular velocity of a cross section of the rod were taken as quasi-velocities to express the state of Kirchhoff elastic rod. The commutative relations of variation and differentiation about quasi-coordinates were derived. The Boltzmann-Hamel equation of a Kirchhoff elastic rod in terms of quasi-coordinates was established according to the Hamilton principle of rod dynamics, from which Lagrange equation can be derived. It is evident that the expanded form of Boltzmann-Hamel equation expressed by quasi-coordinates is Kirchhoff dynamical equation. The Boltzmann-Hamel equation in canonical form was deduced by use of canonical variables and the Hamiltonian was defined. All of the statics of a Kirchhoff elastic rod and the dynamics of a rigid rod are the special cases. A simple example is given to illustrate the application of Boltzmann-Hamel equanon of Kirchhoff elastic rod dynamics.