非对称截面环状Kirchhoff弹性细杆的拓扑构型分析

Topological Configuration of Kirchhoff Thin Elastic Ring Rod with Asymmetric Cross Section

为了揭示非对称截面环状Kirchhoff弹性细杆受到外力(矩)作用时其拓扑状态的变化情况,引入了等效初始扭矩的概念,利用势能密度函数与哈密顿函数形式上的相似性,建立了其与系统哈密顿函数、初始自连接数之间的定量关系.以环状DNA分子为背景,对扭矩模型进行了非线性动力学分析,并利用解析方法中的待定固有频率法,分析了主要参数对扭矩模型稳定构型的影响.最后,利用等效刚度得到相应状态下曲率的表达式,为进一步认识和描述DNA环受到生物酶作用下的稳定构型提供了新的理论途径.

Considering the forces and torques acted on the Kirchhoff thin elastic ring rod with asymmetric cross sec- tion, equivalent initial torque was introduced as a new concept to reveal the quantitative relationship between the state properties and parameters of torque dynamic model, which determines the configuration of the ring rod. As there exists formally a similarity between the potential energy density function and the Hamiltonian function, the quantita- tive relationship was obtained, which connects the equivalent initial torque, Hamiltonian function and initial link number. With DNA ring molecule as the physical background, this paper made the nonlinear dynamic analysis of torque model, and used undecided fundamental frequency method to obtain the asymptotic expressions for steady state periodic solutions around the practical equilibrium point. When the equivalent initial torque took different values, the graph of arc length-torque and phase portrait were drawn. The curvature could be expressed by the torque function with the equivalent bending stiffness considered, which provides a new methodological way to understand and describe the stable topological configuration of DNA ring under the influences of biological enzymes.