CAJ | 학술논문

以脱氧核糖核酸和工程中的细长结构为背景,大变形大范围运动的弹性杆动力学受到关注.将分析力学方法运用到精确Cosserat弹性杆动力学,旨在为前者拓展新的应用领域,为后者提供新的研究方法.基于平面截面假定,在弯扭基础上再计及拉压和剪切变形形成精确Cosserat弹性杆模型.用刚体运动的概念描述弹性杆的变形,导出弹性杆变形和运动的几何关系;在定义截面虚位移及其变分法则的基础上,建立用矢量表达的d’Alembert-Lagrange原理,在线性本构关系下化作分析力学形式,并导出Lagrange方程和Nielsen方程,定义正则变量后化作Hamilton正则方程;对于只在端部受力的弹性杆静力学,导出了将守恒量预先嵌入的Lagrange方程,并讨论了其首次积分.从弹性杆的d’Alembert-Lagrange原理导出积分变分原理,在线性本构关系下化作Hamilton原理.形成的分析力学方法使弹性杆的全部动力学方程具有统一的形式,为弹性杆动力学的对称性和守恒量的研究及其数值计算铺平道路.
이탈양핵당핵산화공정중적세장결구위배경,대변형대범위운동적탄성간동역학수도관주.장분석역학방법운용도정학Cosserat탄성간동역학,지재위전자탁전신적응용영역,위후자제공신적연구방법.기우평면절면가정,재만뉴기출상재계급랍압화전절변형형성정학Cosserat탄성간모형.용강체운동적개념묘술탄성간적변형,도출탄성간변형화운동적궤하관계;재정의절면허위이급기변분법칙적기출상,건립용시량표체적d’Alembert-Lagrange원리,재선성본구관계하화작분석역학형식,병도출Lagrange방정화Nielsen방정,정의정칙변량후화작Hamilton정칙방정;대우지재단부수력적탄성간정역학,도출료장수항량예선감입적Lagrange방정,병토론료기수차적분.종탄성간적d’Alembert-Lagrange원리도출적분변분원리,재선성본구관계하화작Hamilton원리.형성적분석역학방법사탄성간적전부동역학방정구유통일적형식,위탄성간동역학적대칭성화수항량적연구급기수치계산포평도로.
Thin elastic rod mechanics with background of a kind of single molecule such as DNA and other engineering object has entered into a new developing stage. In this paper the vector method of exact Cosserat elastic rod dynamics is transformed into the form of analytical mechanics with the arc length and time as its independent variables, whose aims are to find new tools for studying rod mechanics and to develop the area of applications of classical analytical mechanics. Based on the plane cross-section assumption, a cross-section of the rod is taken as an object. Basic formulas on deformation and motion of the section are given. After defining virtual displacement of a cross-section and its equivalent variation rule, a differential variational principle such as d’Alembert-Lagrange one is established, from which dynamical equations of thin elastic rod are expressed as Lagrange equations or Nielsen equations under the condition of linear elasticity of the rod. For the rod statics when there exist conserved quantities, Lagrange equation which makes use of these quantities is derived and its first integral is discussed. Finally integral variational principle is derived from differential one, and expressed as Hamilton principle under the condition of linear elasticity. Hamilton canonical equations in phase space with 3×6 dimensions are also derived. All of the results have formed the method of analytical mechanics of dynamics of an exact Cosserat elastic rod, so that the further problems such as symmetry and conserved quantities, and numerical simulation of the rod dynamics may be further studied.