超细长弹性杆精确模型的运动学问题

Kinematics of Exact Model for a Super Thin Elastic Rod

作为研究DNA等生物大分子链力学行为的模型,考察弹性细杆精确模型截面随弧坐标和时间的运动学问题。给出了基本假定,用映射的概念阐明离散化思想,得到了杆的运动学方程,导出了存在拉/压变形时弯扭度和角速度的关系;定义了截面的虚位移,表示为弧坐标和时间变分均为零的变分法则,在微分和变分运算次序可以交换的前提下,导出了截面虚角位移的导数与截面弯扭度以及角速度变分的关系,为建立超细长弹性杆精确模型动力学的分析力学方法准备理论基础。

As a mechanical model for studying mechanical behaviors of biomacromolecules such as DNA, the kinematics of a cross section of the super thin elastic rod about arc coordinate and the time were discussed. Giving basic hypothesis of the discussion and expounding the thought of discretization of a rod by maps, the equation of the rod motion was written in terms of generalized coordinates and a relation between curvature-twisting vector and angular velocity of the cross section of the rod was derived. Defining virtual displacements of the cross section of the rod which was expressed as a variational method, in which the variation of the arc coordinate and the time were zero and operation order of differentiation and variation was commutative, the relation between the derivative of virtual angular displacement with respect to arc coordinate （or the time） and curvature-twisting vector （or angular velocity） of the cross section of the rod was derived. The work of the paper may be as a theoretical basis of establishing analytical dynamics method for the exact model of a super thin elastic rod.